I looked, and unfortunately the only removal procedures are for the units with a speaker.
YMMV however, I've noticed no problems having the gps unit facing random directions, such as in front of the heated seat and accessory outlets. I've also had it running in the glove box and it managed to get a signal. May be different for different brands. If the compartment is anything like the speaker section, it will be held in with two screws. Another post included the pieces needed to remove. It's a bit of a process. I took my speaker out to make room for an ipod, and put the grill back on. That might be a good location for a gps. You could also route the wire through the center and splice it into the car somewhere, or simply snake it back to the front and plug it in while hiding most of the wire.Just something to keep in mind before you go through the effort of doing this.
It would look sharp BUT the GPS might not function. The satellite signals the GPS uses are line of sight, so if you mount it to the back of the tray, the receivers that are on the back of the GPS unit (which would normally be facing out the windshield so they get lots of signal), might not be able to receive the signal. I'd test it out with some plastic materials in back of it first, before you do the mod.