It will be a challenge, especially if both the garage wall and the utility room wall are framed with just 2 x 4s. The sink will require a vent line as well as a drain, so getting everything (vent, drain, and supply lines) to fit inside of a 3-1/2" space without seriously weakening the wall studs will be almost impossible. And there's also electrical runs in both walls, so you'll need to work around that as well. Meaning you will need to "bump-out" both wall thicknesses, possibly by adding supplemental 2 x 2 framing (glued and screwed) to the studs after you've pulled the sheetrock.
If it were mine, I'd have a few plumbers come in to give a free quote for the work, enabling you to pick their brains regarding what would work best. You might even consider using one of them to plumb everything, with you buttoning things up after, still saving a bunch of money. Your being in AZ means the house is on a concrete slab (right?), which will make the project that much more difficult. Not impossible, just not as easy as being on a crawl space or basement. Take a few pix and post them here, after you've removed the drywall. And practice up your sweat-soldering skills in the meantime if you plan to do all of the plumbing yourself.