I'll answer my own question since I spent last weekend up there (pictures and hiking logs to follow). For anybody looking to "car camp", the Pinalenos don't offer many options outside of developed campgrounds. I actually sleep in the back of my truck most of the time, and that's not even possible at the Hospital Flat Campground, where you have to park in one area, and set up camp a short walk away.
Nonetheless, there are probably about a dozen spots off on one of several 4x4 roads that are suitable for camping and solitude with vehicular access. Don't count on finding them empty unless you're there early though, as I found most occupied by Friday mid-afternoon.
As stated above however, I found numerous great spots for dispersed camping while hiking the trails. That seems to be the way to go when spending the night on Mt. Graham.
On a positive note, presumably due to the lack of roads and 4x4 trails up there, there's not a lot of traffic. So you won't be frequently disturbed by Quads, 4x4, or motorcycles cruising past your serene wilderness experience (as frequently occurs along the Mogollon Rim and White Mtns.)