From DwarfFortressWiki

Quests in adventure mode can be obtained from leaders in settlements (mayors or civilization leaders). Civilization leaders will give harder quests than mayors.

Quests can be:

• Killing a hostile creature at a particular cave, ruin, camp, etc.

Rewards can be:

• Having your character's kill recorded in legends.
• After many kills of those creatures your character will earn a title of good, if he doesn't already have a title already.
• You character will be affiliated with the entity you solve the quest (town or civilization)
• Other than that, nothing else that'll benefit your character besides the experience gained from the kill.

[edit] Finding a quest location

The location of a quest can be found quite easily on the Travel screen. Once in the location, you can use the minimap to know the general direction of the feature entrance.