So both side quests and Librarian, Bounty, and Hunter quests all count. For those latter you must actually go turn in the item/bounty/trophy for it to count.
There’s enough available side quests to earn this by the end of The Quarry area (this will involve back tracking as you don’t get access to hunts and bounties until Trench Town and there are some available in earlier areas).
You can identify which areas have available side quests of any kind by viewing the travel map (just talk to your ultra cuddly Polish pal by the truck). Exclamation marks, paw prints, target reticule, and a book mean there’s an uncompleted quest in that area while a checkmark means there’s one that should be turned in there.
Note: there’s at least one, long multi step side quest that takes you through multiple areas, I’m unsure of whether each step counts as a quest or if it must be completed all the way through to count. Regardless, I had the Hunt mission in The Quarry bug out and still was able to get this from quests completed in that and previous areas only.