We have the Furrion Observation Camera, very happy with it an took all of 15 minutes to connect and get working.
To answer your question, yes you can use the Garmin camera using the supplied Furrion wiring, the only thing inside the Furrion black box is the 12v power for the camera. On our WJ the supplied 12v power is connected to the driving lights. So to get power to the camera, the trailer umbilical cable needs to be connected to the truck and the driving lights turned on.
The Garmin RV760 GPS has a wired camera connection and then the Garmin BC-20 camera is a wireless system that includes the wireless TXer/RXer. The wireless RXer connects into the Garmin wired plug. You can buy a Garmin camera cable for ~$20 or less and use any wireless setup.