At the moment, there are no quests available within Project Zomboid ever since the removal of the Baldspot & Kate tutorial, 'Til Death Us Do Part.
The closest thing to quests at the moment is the inclusion of annotated maps, which can be found by chance on Zombie corpses. An annotated map can contain directions to hidden stashes, similarly to a treasure map.
Quests are planned to be added as development continues. However, the people, items and circumstances involved with quests are suspected to be different each time a new game is started.
Future possibilities include: the player encountering random NPCs that need Items, player may have lost relatives, player may be able to trade information for a stash of goods, and more.