To directly answer your question, I don't think there is a safe way to haul a full size quad on an ATV carrier just attached to your existing hitch. I think a quad would be way too heavy to be safe on that type of mount. But as already mentioned there are ways to do it, but it would involve welding and probably a three point attachment to the motorhome to do it safely.
Now other things to consider, I haul my quad in the bed of my Dakota toad. I don't know where you plan to camp, but I would think the vast majority of campgrounds usually prohibit the use of off road vehicles (such as quads) within the campground. There are many campgrounds that specifically do allow the use of quads and they are often adjacent to off road use areas.