Assuming you mean like D&D, kind of.

Greek Heros, such as Perseus, Hercules, and Jason (his Argonauts would have been his party!) They had quests, they fought monsters, gods, titans, and evil kings. They found magic items, they received rewards, ect.

Western Europe had two versions, the charismatic outlaw, and the knight errant (lordless knight). In popular culture for the day, both were heavily romanticized for having adventures, going on quests, finding relics, and such.

Japan had ronin, and the idea of Musha Shuga (sp), or warrior pilgrimage. Ronin were like knights errant, in that they didn't have a lord. They wandered the land as swords for hire. A musha shuga is when a samurai went on 'holiday' to wander and learn. It could be a religious trip, or it could be educational, or it could be (mostly in stories) "I go challenge the sensai of every dojo I run across to a duel!". The idea of the wandering duelist falls into this, with people like Musashi.

China I'm less familiar with, but I have heard stories about wandering magistrates. They roam the land acting like a less murderous Judge Dread, using their martial skills and wisdom to subdue villains, and judge crimes. Also wandering monks and wisemen, ect. And last but not least, you have "warrior friends" stories, such as seen at the start of Romance of the Three Kingdoms, where a group of young men band together to fight an "evil" enemy, such as an immoral rebellion. Chinese myth also has such villains as "the Eunuch"

So, this isn't an exhaustive list, as I've left out Biblical heroes, Persian, India, Russian, and so many more, but it fits a pattern.

To bring this back to your original question, yes there is evidence that people tried to live these lives, though most of what I've read about it is more about them trying to live up to the stories, rather then being the basis of them. The knights errant for example. Being a knight is expensive, with the armor, weapons, horses, and retainers. So to go on a quest, you need money to support you. Unlike D&D, we don't have random dungeons full of treasure, guarded by a few orcs. Any cash on hand (or equivalent value in goods) is owned by someone, so to get it, you're either killing and robbing them (not very heroic), or you end up going mercenary (also not very heroic, but more moral, even if just by a tiny amount) if you want to keep "adventuring".

Hi. You might consider x-posting this one to /r/AskLiteraryStudies, who could give you a survey of western literary tradition (e.g. ancient epics, Norse sagas, Arthurian legends, Don Quixote, and so on), but since you ask about historical reality, you may find these posts of interest:

• /u/Rittermeister in In games and books there are always "adventurers" who do odd jobs and explore for a living. Did something like this ever actually exist in medieval times?. As an aside, I see that I pasted a block of links to threads about mercenaries in there; no doubt those links should be pruned aggressively by now, so just a heads up that some threads may not pass current subreddit standards.

• /u/AncientHistory in Was "adventurer" (in the Dungeons and Dragons sense) ever an actual profession in medieval Europe?

• /u/restricteddata in Did 'brave adventurers' actually exist during the middle ages, or is it just a videogame trope?