Quests
This feature has been removed from the game.
It has been replaced by something else.
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.
Maps
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.
Future
This is a future feature.
This article is about an item or mechanic that is currently unimplemented and planned for a future version.
It has either been previewed or found in the game files and does not exist yet.
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.