There are a total of 8 Quarantine Zones in Dying Light. Completing all of them will earn you the "Trespassing" trophy or achievement. By default these zones will not show up on your map. Only when you get close to them they will be marked.
Most of the Quarantines require you to pick up some supply crates in buildings or parking garages. In the others you have to kill all enemies to clear the area.
Before attempting to clear the Quarantine Zones you should complete the story and get some good weapons. The reason being that these zones are overrun by hordes of zombies and you won't stand a chance if your weapons aren't up for the task.
You can also do this in Co-Op to make things easier.