I've been working with Quinn in every spot since her release. Since her release the jungle has change a large number of times. As of patch 4.11 the jungle machete has changed notably and now grants +5 damage reduction vs. monsters and +10 attack damage vs. monsters. This greatly benefits junglers with naturally low armor and who are reliant on AA. Likewise the Spirit Stone and Spirit of the Elder Lizard changed in patch 4.3 and now grant BUTCHER: Against monsters, deal 20% bonus damage and restore 6% of damage dealt to monsters as health and 3% as mana. Butcher allows Quinn to maintain her health AND mana while clearing the jungle. Finally Spirit of the Elder Lizard has INCINERATE: Deals an additional 14 + (2 × level) bonus true damage over 3 seconds upon dealing physical damage. This is important to note because Quinns Q - Blinding Assault procs Incinerate on ALL targets. This greatly increases Quinns ability to quickly clear the jungle and makes her ganks more dangerous for all enemies as they will have trailing true damage on them as they fight and flee (which helps secure kills).

With the new Machete and current Spirit of the Elder Lizard Quinn works well in the jungle playing much more safely than she did pre-6 and now having strong post-6 ganks, heavy late game damage, and extreme mid game split pushes.

I've been playing her in the Team Builder and mopping up carrying nearly every game, crushing opponents and tearing their teams to shreds. Given enough time to get her core she will quickly out level and out power enemies becoming an dangerous character to approach.

The key to success with Quinn is how you build her to adjust for the jungle. And understanding how to properly play her in order to get your team fed.

So what can you expect jungling with Quinn?
A) You will have a somewhat slower start, your first jungle pass will be slow till you have your spirit stone. With this said the new machete works great on her and makes jungling fairly safe early.
B) You have to watch carefully for counter jungling and need to ward. Very fast junglers like Shyvana, Lee and Shaco will be dangerous. Specifically you'll want to watch for team counter jungling where bottom / mid heads into the jungle after you.
C) Despite your slow start you can often push for a team kill around level ~3 if one of your enemies has pushed and your teammate is paying attention. Be careful to NOT get caught by the enemy at the same time as you are squishy early.
D) Once you hit level 6, if you're not behind you can start ult diving lanes, get ganks and start snowballing.
E) With Spirit of the Elder Lizard your jungle clearing will go quickly, and with Ionian Boots of Lucidity (and the recommended runes/masteries) you should have near max CDR allowing you to use your ultimate approximately ever minute (84/66/48 seconds) this combination will allow you to gank extremely frequently with speeds no other jungler can match, putting any enemy with half or less hitpoints (or simply to far from their team) at danger from being assassinated or ganked by you. This makes your presence felt by everyone on the map.

Specifically for ganks you want to watch each lane. When you see someone damaged and mid or farther in, you want to quickly head there, pop your ult EARLY before they can see you and sometimes super early to reach them as quickly as possible. Remember your ult has a massive movement speed boost out of combat. Consequently you can use it, dive toward someone damaged, pop E to ponce them, pop q, and chase till dead. This works so freaking well its not funny. If your ult is up, you are a serious danger to any opponent at half health, because you can traverse the map so quickly and lock them down when you get there.

Jungle Quinn is very viable right now, due to her skill set. And with the recent jungle change her power will grow over the game with healthy gold income as if she is playing top or carrying heavily.

Here are my thoughts on skill priority. I rarely find reason to deviate from the below.

R > Q > E > W is generally what I do for jungle. Obviously you want your ultimate as beefy and often as possible. After that you need the as much Q damage for quick clearing. Next you want your pounce ability E to be on a shorter cooldown timer (and more damage).