Other classes have easy explanations for how they came to be: fighters and monks trained, wizards studied, clerics and paladins prayed, sorcerers and oracles were born that way, and so on, so forth. Druids, however, do not seem to have any explanation for how, exactly, they became druids.
The pathfinder druid page gives the following class description (emphasis mine):
Within the purity of the elements and the order of the wilds lingers a power beyond the marvels of civilization. Furtive yet undeniable, these primal magics are guarded over by servants of philosophical balance known as druids. Allies to beasts and manipulators of nature, these often misunderstood protectors of the wild strive to shield their lands from all who would threaten them and prove the might of the wilds to those who lock themselves behind city walls. Rewarded for their devotion with incredible powers, druids gain unparalleled shape-shifting abilities, the companionship of mighty beasts, and the power to call upon nature’s wrath. The mightiest temper powers akin to storms, earthquakes, and volcanoes with primeval wisdom long abandoned and forgotten by civilization.
This seems to imply they are granted powers in a manner similar to clerics, but then the same page goes on to say:
Druids worship personifications of elemental forces, natural powers, or nature itself. Typically this means devotion to a nature deity, though druids are just as likely to revere vague spirits, animalistic demigods, or even specific awe-inspiring natural wonders.
Since druids who worship the concept of nature itself (or some aspect thereof) are just as prevalent as druids who worship any specific entity who may be acting as their patron, it seems more as if they are drawing power from nature, rather than being granted power. The overall druid fluff seems pretty clear on the idea that the amount of druidic power you have access to is based on your affinity with nature, (represented by character level) which explains why ceasing to revere nature causes you to lose that power.
The 9th level spell World Wave supports the "oneness with nature" concept, noting that:
creatures with even one druid class level (regardless of their type), are considered a part of the natural world
But the question remains of how one gains that first druid level in the first place. Do you just live as a hermit until you become one with nature and the powers develop on their own? Do you seek out and learn the druidic arts from an existing druid? On any race page, you can see druid is listed under the "Trained" age range, alongside alchemists, clerics, inquisitors, magi, monks, and wizards, which implies it is a time-consuming process to become a druid.
Mechanics-wise, it is perfectly feasible for, say, a tree-hugging monk to multiclass to druid, so there doesn't seem to be much of a barrier for entry into the class, beyond the characteristic devotion to nature, unless this is a handwave for player convenience.
Is there any official Paizo content that describes how a non-druid becomes a druid?
In the absence of Paizo content, I would also accept an answer based on D&D 3.x content, so long as it was still consistent with pathfinder lore.
A 3.5 perspective - the background section of Druid in Player's Handbook:
Player's Handbook 2 gives another bit of detail:
One more clue, and possibly the biggest one, is the druidic language (which remains in Pathfinder with similar restrictions). No druid can teach it to non-druids, yet all druids know it. While I cannot comment on whether this druidic society exists in Pathfinder, at least the language part must be taught by another druid. Though, generally, I'd guess a druid is similarly trained by other druids. Keep in mind this is all subject to change according to the DM's views of the setting.
As to becoming one in the game, mechanics-wise, there shouldn't be any restrictions, so long as you meet the alignment requirements. One would assume that you found a druid and got inducted into the society "off the screen". It should not be hard to pass any tests when you have character levels, and the society would be happy to have a competent and resourceful member. It is probably similar for quite a few of the classes. After all, if you multiclass as a wizard, you probably did not just buy a blank spellbook and reinvent magic?
The main consideration here is probably the convenience and enjoyment of other players - this is most likely a solo adventure, and everyone else would be just twiddling their thumbs bored.
The distinction between drawing power and being granted power is very blurry, opinion based, and subject to DM's and everyone else's interpretation of deities and the divine. Additionally, 3.5 similarly allows clerics to be dedicated to a cause, alignment or idea (and glosses over on how the non-deity clerics come to be), just as druids can be dedicated to nature, which may imply the existence of some differences.
With that out of the way, I would doubt clerics are "granted" power, except in the interpretation where gods appear to you personally or operate call centers. It's probably more a matter of establishing a divine connection, and then drawing/calling on that connection to grant you power/miracles. Which would be rather similar to druids and their connection to nature. I'd guess the "rewarded for their devotion" quote should be interpreted less directly - nobody specifically rewarded you with the power, the consequence of the process of devoting yourself to nature is access to the power of nature. The druid class then provides the combination of a connection to nature and knowledge on how to channel the power of that connection.