"A knight goes on a quest" is mostly a trope of fiction. But that trope will often conjure imagery of slaying a dragon or monster.
Is it not possible that one from Republican Rome could travel to Egypt and slay a Nile Crocodile? If I were a Roman patrician I'd be pretty impressed if someone slayed a crocodile. For someone predisposed to enjoy hunting, I'm sure that would be fun. There was an enormous tradition of hunting tigers during the Raj in India, but before that? It isn't too big of a stretch to tell a realistic story involving a travelling hero saving a remote village from a monster, if that monster is a tiger with a broken jaw, leading to man eating tendencies. This trope can be seen in modern story telling, from RPGs to hollywood movies, but is common in mythology the world over, in unconnected canons like English and Chinese mythology (Rescue a maiden from a monster can be seen all over).
So was there any tradition or individual incidents of people slaying "monsters" or big predators as an occupation or even hobby, in medieval, or preferably pre medieval periods?
Many thanks.
Across many cultures, hunting was seen as a Big Deal(tm). It tended to be primarily the purview of kings and noblemen-- people with the free time to go hunting, and the lands to go hunting on. Killing stuff was seen as a good way to prove how manly you were, and all that.
This said, a king or nobleman going out on a hunting trip was more akin to an army going out on campaign than a lone knight going on a quest. He'd go with dogs, servants, 'beaters' to surround the quarry and make lots of noise to flush it out, and so on. Think less a guy in a deer stand, and more the whole rigamarole of an English style Fox Hunt.
Probably the closest thing to a 'monster' hunt would be boar hunting. A wild boar is several hundred of pounds of tusks, muscle, and hate, and as such hunting them could be pretty dangerous. The 'traditional' way to hunt boar was to goad the thing into charging you, then skewer it with a specialized 'boar spear' when it did. The spear had crossguards just past the blade to (theoretically) keep the boar from charging all the way up the haft to gore you. And they did that kinda thing for fun.
So that covers stuff for hobby-hunting. As for something on a more professional level, many European kingdoms had policies in place to exterminate wolves-- presumably there were people who made a living at it. The French even had a specific title of "Wolfcatcher Royal," which was supposedly the descendant of similar wolf hunters who worked under Charlemagne.