It depends which quest you have exactly, but many are Explore quests; basically you just need to explore - in any biome - until you find the relevant encounter. (Edit to add: and ofc you must then click the correct option on that encounter, e.g. the 'tend wounds' button for the injured lioness quest, or 'attack' for the various battle quests you have to win)

The wiki section here lists all the quests and how to fulfil them, including links to images of the appropriate explore encounters if you're not sure what they will look like :)

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