The problem I see with this is that once a pet is entered into a quest, it's zapped of all it's energy anyways. Canceling the quest would still leave you with an energy-less pet, which renders it useless for a time. Even if you suggest quitting a quest will give you back your energy, that doesn't seem right. I see it going like this; Someone decides to enter a pet in a quest. On a whim, they want to go exploring but have no pets with energy. That's not a problem, just quit a quest and have another pet to explore with again.
I think of it like battling while you're exploring. Whether you lose or quit, the two outcomes force you to leave the island. In this respect, I can't support a quest quitting option unless it still leaves your pet unusable for explore/other tasks.
My thoughts on this could change when tomorrow rolls around, as I want to test something. As of now though, no support from me.
"You know what the issue is with this world? Everyone wants a magical solution for their problem, and everyone refuses to believe in magic."
- Jefferson/The Mad Hatter, Once Upon a Time