If you plan to visit Quneitra, you have to get a permission from the Ministry of the Interior in Damascus. This can be found at Al Malki Square. Normally it takes no longer than 10 min. to get the permission, but keep in mind that the office is open Su-Th only from 8am to 2pm. You can only get a permission for the current or next day. Don't forget to tell the officer on which day you want to go.
Hitchhiking within and out of Quneitra is definitely not possible, because you'll have to go around in a minibus, taxi or on a motorbike together with a soldier at least to the next city.
It is possible to hitch to the village just before Quneitra. From there, several UN checkpoints will be on your way and walking through is not possible. You can try hitching a UN car or maybe some of the few locals will bring you to the last checkpoint, where a soldier will join your further trip. This one is located 4km before the city, and walking there is not allowed, so you have to take a minibus or taxi. If you're lucky, the army guys will give you a cheap tour (100 SYP) through town on their motorbikes.