quail are going to hang out near dry washes, etc. So anywhere out of town in areas like that. The closer to town, the more people are going to hunt there. I used to go down by Ajo and get in to a bunch of quail there. Been a couple of years since I was there though. 4.5 hours of travel time? I think that will get you out of state in each direction. I am from Yuma originally, so go back down that way for dove, quail, and ducks.
As for deer, AZ has a drawing system, and it closed in June. The drawing was on 7/8, and it will be posted later this month. There are lots of places to hunt deer. I put in for the Ashfork Williams areas myself.