There are many good answers with general advice already, instead, I will show you what the game rules advise on how you should be handling this since the core rulebook is vague about social conflicts in general.
Then, it describe many possible scenarios of characters using Bluff that could (and will) come up during play, be that on social conflicts or not:
This section includes clarifications and details on several different ways to use Bluff—and on several things that don’t work. (...)
Bluff Doesn’t Define a Response: Even the most successful lie told using Bluff doesn’t determine the course of action the deceived person takes—it just primes the target with misinformation.
This means attempts to trick a creature into a course of action might need to also include Diplomacy or Intimidate after the Bluff check. For example, suppose there was a guard with the following orders from the guard captain: “Don’t let anyone into the restricted area without clearance papers, even if it seems to be me or someone of higher rank.” After this, a sneaky rogue attempts the following ruse: “I am the king’s general on a mission of utmost importance for national security. I need you to let me in now, or you’re fired!” Assuming the rogue succeeds at her Bluff check, the guard now believes her to be the general, but this doesn’t mean he will let her through.
His orders still require him to keep everyone out without papers. The last part of the rogue’s demand is an attempt to Intimidate the guard, and the successful Bluff check was a necessary prerequisite to even attempt the Intimidate check.
Circumstances: When using Bluff to tell a lie, the Core Rulebook table on possible circumstance modifiers takes into account several levels of plausibility, targets who want to believe or are impaired, and possession of convincing proof, but there are also plenty of other circumstances that might affect the result of a Bluff check. For instance, many people strongly don’t want to believe a bluff that would lead to cognitive dissonances, such as attempting to convince a true believer that their religion is fake, and such a lie imposes a –5 penalty on the attempt (the opposite of a target who wants to believe the falsehood).
On the other hand, a target who is afraid that the deceit is actually correct might grant a +2 bonus or more on the skill check, depending on the level of anxiety about the fabrication. For instance, a bigoted assassin who is afraid that half-orcs are cannibals might be more likely to believe a half-orc’s bluff that she ate the target he was supposed to kill.
There are a variety of other circumstances, all of which might alter the odds in different directions. A character with a widespread reputation of being a compulsive liar might take a large penalty on his skill check, but a character with a reputation for always telling the truth, such as a paladin, would gain a large bonus on her skill check. Similarly, a hostile creature is much less likely to believe a deception, whereas a helpful creature is much more likely to believe one.
Tricking Someone: Bluff can be used to cleverly trip a target up and get him to reveal something or make a mistake. In these cases, he realizes his mistake soon after, but by then it is too late, and the falsehood has done its damage. This is similar to using Bluff to feint or create a distraction, but has broader applications in social situations.
For instance, suppose a swashbuckler suspected that an assassin works for the queen. The swashbuckler might be able to trick the assassin into revealing more information by pretending to be a fellow agent of the queen in an attempt to gauge the assassin’s response. Of course, if the assassin doesn’t work for the queen and sees through the ruse, he might attempt his own Bluff check to pretend that he works for the queen and fell for the trick, thus causing the swashbuckler to investigate the innocent queen.
Plausibility: The Core Rulebook mentions that some lies are implausible enough that no matter how high a character’s Bluff check, a PC can’t convince a target that they are true. However, the same page also presents a table that says that “impossible” lies impart a –20 penalty on the skill check. This table’s entry might actually be better described as “particularly implausible.” For example, an older human woman telling a very similar-looking human girl that she is herself from the future might take the –20 penalty, whereas a 10-year-old half-elf telling a 40-year-old orc the same lie would automatically fail the Bluff check.
You’re Not Lying, You’re Just Wrong: Sometimes a character is a convincing enough liar that targets can’t tell the character is lying, even when the targets possess incontrovertible proof that what the character is saying isn’t true, or the lie is otherwise too unbelievable to be possible. In this case, one way to resolve the situation is for the bluffing character to take a –20 penalty on the skill check, and if she beats the target’s Sense Motive, then the target believes that the bluffing character isn’t lying, but is simply mistaken.
This could also be the result of other situations in which the target of the Bluff attempt has strong reasons to believe that the falsehood, despite being plausible, isn’t factually correct. Even this result can be useful to the bluffing character, as it doesn’t mark her as a liar, and it allows her to gather information about what her target knows and expects.
True Lies and Implausible Truths: Bluff is the skill that convinces someone that something is true. However, there are a few potential cases when the situation isn’t as straightforward as a bluffing character telling a lie to a target.
For instance, suppose that the bluffing character makes up a believable lie to tell the target, and the lie turns out to be true, unbeknown to the bluffing character. If the Bluff check succeeded, the target is convinced, and might later verify the truth and trust the bluffing character more. However, what if the bluffing character fails? In this case, the target can tell that the bluffing character is lying, but that doesn’t necessarily mean that the target is forced to conclude that the information is false.
For instance, suppose a popular king has fallen into a magical, unbreakable sleep. A charlatan, noticing the king’s lack of public appearances, makes up a story about the king being placed under a sleeping curse and spreads it around the tavern, but his Bluff check is terrible, and everyone can tell he’s making it up. If one of the king’s advisors is present in the tavern, this doesn’t mean that the advisor now thinks the king isn’t in a coma; it just means that she can tell the charlatan doesn’t believe his own story.
The reverse side of true lies is implausible truths. These are situations in which someone is telling the truth (either saying something that is actually true, or spreading a lie that they believe to be true), but that truth is extremely implausible to the listener. Though the bluffing character isn’t lying, the same skill set that makes an excellent and convincing liar could potentially help characters attempting to spread an implausible truth.
In these cases, even if the target succeeds at the Sense Motive check, he can tell that the bluffing character truly believes what she is saying, and he might simply conclude that she isn’t lying, but simply mistaken. The target might later be swayed if presented with evidence or through a verbal duel (see pages 176–181). If a bluffing character successfully convinces a target of a lie and the target attempts to spread that information, this leads to a classic example of an implausible truth.
Personally, I never roll for Bluff when GMing NPC's (unless its being used in combat), I use 10 + Bluff bonus as the DC of the PC's Sense Motive check. Those checks may be asked on demand by the players, and are rolled in secret (even if the NPC is not actually bluffing). Also be affirmative on the result: "She is speaking the truth" and not "She appears to be speaking the truth". Unless your intent is exactly to get them on the edge and suspicious of everybody they meet (which is fine for a horror game).
When discovering this bluff may be important to the history and has roleplay potential, I will do the opposite and roll all players Sense Motives against the bluffer (also in secret), using 10 + Sense Motive bonus as the DC for that check.
If one of the PCs pass on that check, I will inform that the PC(s) notice that something is not right in what they are being told, but not exactly what, they will have to roleplay their way out of it and possibly make more Sense Motive checks, along with some Bluff and Diplomacy checks. The NPC may simply be hiding something, they may be afraid of being discovered, or are afraid of their boss, or are sad or angry about something they just mentioned, and so on. They have something to hide and that something could be discovered by the players, so a Bluff/Sense Motive check is warranted.
Best Answer
Let me start with a similar statement:
"Another PC came up to me, and killed my character in one shot. I didn't like it, so I said he missed."
In many ways, that above statement is identical in kind, if not in specific consequences to the one you proposed. The fundamental resolution mechanic is that the defending player in this or in the social setting has no way of demonstrating agency over her character. This removal of control is either explicitly or implicitly fought because it goes against the patterns established by the other methods of conflict resolution in the game.
Vincent states:
This scenario fails due to the latter component, there is no compromise or mutual partial success, as there would be in a combat between the two characters.
If we consider the area of cRPGs, we get Player Skill v. Character skill articulated in an interesting way here:
...
Now, while he's talking about cRPGs, there is an interesting conclusion to be drawn about typical character-conflict-resolution mechanics. Most games social contracts have a "Don't roll dice against other players" provision, allowing for player persuasion against player. By allowing dice rolls, without the adjudicating components of the NPC game world informed by the GM, there is a one-hit win/loss and the game really does become a sort of progress quest for that interaction. All agency is removed and the losing party feels powerless.
There are three games with absolutely fascinating player conflict resolution mechanics. DitV, Mouseguard, and Ars Magica.
DitV articulates all conflict in one conflict-system, from talking to gun-fightin'. Therefore, if players want to engage in what is PvP conflict, there are ways to interact that provide both sides agency in getting what they want (as well as the ability to roll fun handfuls of dice.) One bad roll won't make someone feel powerless, and both sides can impact the others' reality (do damage) before one ultimately triumphs. Furthermore, there is a mix of player and character skill, as the conflict is rendered both mechanically and narratively. Both sides buy into the rules because they realize that the rules will fairly impose unwelcome outcomes on both sides.
Mouseguard has a similar philosophy, providing an argument subsystem and raising player ideological conflict to a fairly important role. Therefore, players playing mouseguard realize what will happen, and buy into the rules making unwelcome situations for them. Here, there are explicit elements of compromise baked into the system and both player and character skill come into play.
Once again contrast this with kids playing "cops and robbers" wherein one says "bang bang, I shoot you" and the other says "no you didn't." Without mutual buy-in on the rules, this is what a RPG devolves to. Very few people will willingly choose to buy into a lack of agency on their behalf.
Both of the above games can be characterized as narrativistic. However, Ars Magica (a decidedly non-narrative game, depending on how a group plays it) also has a mechanic. Certamen, a non-lethal fight of magic, is designed to provide magi within the game a way to compel action by other magi. It is a fascinating tool for PCs to compel action or inaction by other PCs. While the concept of a magical duel isn't normally perceived as a social skill, it fits the exact same mechanical niche: a way to compel action with mutual buy-in from all involved. This rule subsystem is also extended to debates, offering a way to have players and characters offer points and counterpoints.
Looking at D&D, unfortunately, the lavish attention they put into the tactical combat system is not reciprocated within skills. While skills are a useful shortcut for impacting the GM's world, the abstractions they use do not provide any provision for player response. Some groups I've played in solved this problem by refusing the rules' right to adjudicate inter-player conflicts, and others turned it into a simple attack: skill versus relevant defense.
Neither solution is particularly satisfying. By refusing the rules, players functionally refuse the interesting aspects of entering into unwelcome situations. By resolving the compulsion as attack, there is no symmetry of unwelcome possibility, nor any way of using player skill to impact the outcome.
This, ultimately, is something that must be decided by your group. However, if the group likes the idea of inter-player conflict, it is probably worthwhile importing expanded and useful rules from another RPG to satisfy the dramatic/situational needs of the table.
It sounds like, in this specific instance, it's worth performing a debrief after the end of the game and assessing what the player desires are. Assuming that this was a big deal to both players, I would make up a simple "social combat" system on the spot, importing the tactical richness of the dominant conflict system to this important PvP conflict.