Jungle is the best role for quinn. Why is this you may ask? Well I'll explain.
In our current meta, quinn is one of those champions that are very powerful, but they do not fit into the meta very well. Quinn isn't a good ADC because her ultimate forces her to get into melee range, while on the other hand she cannot 1v1 beefy top laners such as renekton and jax. In the jungle however, she works incredibly well.

Quinns versatile kit allows her to be a very scary jungler when played correctly. Her passive is just badass. In the jungle, in ganks, or in teamfights, it provides crazy damage and procs her movement/atk speed steroid on her w. Quinns Q, which is a skillshot that explodes into an AOE blind, does a lot for a jungler. It does a lot of damage to jungle camps and blocks a lot of the damage thanks to its blind. As far as ganks go, It both does strong burst damage and disables the opponent to turn around and return damage. Also, quinns Q is insane for counter ganks. An enemy getting blinded under your turret is almost certainly a secured kill. Heightened Senses in pretty OP in the jungle. The active allows massive vision, and the radius is increased the more you level it. It guarantees vision when wards are not an option, and will save your more than you expect.
Vault is your main tool for ganking and escaping fights. Its a 2 way ability. It can bring you closer to an enemy, or push you farther away, depending on your current location. Its your bread and butter skill for jungling quinn
ULTIMATE TIME: your ult is your valor form. This is what jungle quinn is all about. Your ult allows you to be everywhere at once, and makes you incredibly mobile. Ganks are more accessible, objectives are easier to reach, and ganks are easier to stop. On top of all that, it aids in chasing and running away from enemies.

Strengths:
+ Good jungle clear
+ Strong Ganks
+ Lots of utility
+ High Mobility
+ Vision
+ Powerful Counter Ganks

Weaknesses:
- Squishy without items
- Clears slower than meta junglers (Nocturne, Yi, WW)
- Item Dependent