If you're dead set on mounting this inside the garage, a simple solution is to cut a piece of plywood (3/4" should do) about 2" wider and taller than that plastic mounting bracket. Attach the plywood to a stud using some lag bolts with the heads of the lag bolts lined up in the center of the stud and ensuring that vertically, they fall into the spaces in the back of the bracket, roughly where the green circles are.
Once your plywood is lagged to the stud, use the mounting screws included with the bracket to attach the bracket to the plywood. The screws should hold just fine in the 3/4" plywood and if they happen to go through the plywood and extend into the drywall behind, it won't matter. That won't do any more significant damage to the drywall, nor will it help or hinder their holding power.
You could, of course, paint the plywood to match the wall color if you'd like to make it less obvious.
I'll reiterate the advice of others, though, to mount this outside in case there are leaks as it's always preferable to have leaking water outside rather than inside.