Yes, as long as the cantrip has a casting time of 1 Action
Most cantrips require an action to cast. As such, you can certainly quicken them. So any 1 Action casting time cantrip will be usable, such as Eldritch Blast in this question.
However, PHB p. 202 states:
A spell cast with a bonus action is especially swift. You
must use a bonus action on your turn to cast the spell,
provided that you haven't already taken a bonus action
this turn. You can't cast another spell during the same
turn, except for a cantrip with a casting time of 1 action.
Thankfully, if you are planning on casting two cantrips per turn, you are more than able to do this effectively, as they are not restricted by the same restriction as regular spells.
No, you can't cast it using a spell slot.
Spells you can cast because of your race aren't spells you know, and thus can't be spells you prepare. Per the PHB (p. 201, "Known and Prepared Spells"):
Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind ...
... and it goes on to describe some of the class-specific variations of knowing and preparing spells. But in any case, having a feature from your race or class that allows you to cast a spell isn't the same as knowing the spell.
This is spelled out clearly in the multiclassing rules for Pact Magic quoted in the question (PHB p. 164; emphasis mine).
If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.
Likewise, the Drow Magic trait clearly distinguishes the known cantrip from the merely castable other spells (PHB, p. 24; bold for emphasis mine):
You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.
If the intent of the rules was that Drow also knew the non-cantrip spells, it would say so.
Best Answer
No you cannot (With a potential loophole for Warlocks)
Each class spellcasting description specifically describes what its slots can be used to cast.
The Wizard, Bard, Druid, Cleric, Eldritch Knight, Arcane Trickster, Sorcerer and Ranger classes all say:
In these cases, the class feature actively prevents you from casting anything other than 1st level spells or higher with your slots. Since cantrips are 0th level, you can't use one of these slots to upcast a cantrip because these class feature rules are more specific than the general rules on spellcasting.
But what about a multiclasses character?
The multiclass rules say:
meaning that even when you multiclass and cast a spell, you are still limited by the spellcasting feature as defined by the class that it's associated with, with the exception of the number of spell slots you have.
Okay, so how about paladins?
The Paladin spellcasting section and Warlock pact magic feature say:
Paladins don't have cantrips normally, so the only way to get one RAW would involve multiclassing (which we know wouldn't work), or the magic initiate feat. However, the magic initiate feat associates the learned cantrips with a different class. And since the paladin's spellcasting feature explicitly only gives you slots to cast paladin spells, you couldn't use them on a cantrip.
What about Warlocks then?
Technically the pact magic feature doesn't explicitly say that you can't cast a warlock cantrip at a higher level. But there really isn't a point to it. No cantrip to date explicitly says that it has any extra effects what cast with a spell slot, and it is a spell's definition that tells you how the spell reacts to higher level casting. Additionally, the concept of allowing a certain action because there is no rule specifically against it might not sit well with some DMs, as it can open up some doors to some...unintended consequences.
Additionally, Jeremy Crawford, the lead rules designer, has shed some light on this topic saying:
Even if you don't think JC's word is law, it does provide some insight on designer intent. So, even though a DM might technically, with a liberal ruling, allow cantrips for warlocks to be cast using spell slots, doing so is counter to designer intent, doesn't provide any defined benefit, and isn't a sure thing since it relies on a method of reading rules that actively exploits loopholes.