Well I'm sure you could, depends on how thick it is. Perhaps a stock extended cover. I don't know about the extended battery though. It's pushing stock cases as it is.
I'm not 100% sold on the USB port and flex cable. I guess if you use a TPU case, this would mostly be concealed but I've seen these where they actually attach between the battery and battery contact inside the phone (double sided flex cable with copper connections on both side). That would be ideal and I'm sure it would work.
Similar to this one:
It just bends to go in between the battery and phone contacts. These are also meant to go under the battery cover. Our batteries use 4 prongs, not sure if this one listed is compatible. I would buy one in a heartbeat if someone posted a link to one that works.
My N7 is Qi capable and I have a Qi Charger, but it goes largely unused. I'd use this in my bed room though for night charging. I'd like to keep the USB port open still for charging normally in my car or when traveling like normal.
I don't see any actual easy way to do this. The one I posted before actually uses battery contacts under the battery cover that is specifically for this. The Galaxy Nexus does NOT have these. However with a little engineering, it should be just as easy. The battery has 4 contacts. The battery is labeled with (+) and (-) pins. One could probably grab one of these for the Galaxy S3, turn it sideways so the contacts are close to the battery pins, and use some kind of copper foil to wrap around the battery tabs and to the Coil leads.
I use an extended battery, which leaves me with little room. I have been wanting to experiment with cutting one of my backplates in half to just cover the top portion around the camera, giving me some additional room for stuff like this. Probably wouldn't work unless I used one of my stock standard batteries. Losing a few mAh for Qi might be a good trade off though.
The mod would actually adhere to the battery itself, thus no physical modifications are done to the phone. So I could always pick one of these up, hack a battery, and test it.