2005 Nissan Quest AC

It sound like you may have an issue with the rear air condition system.
Do you hear any clicking noise coming from the rear inside the vehicle when adjusting rear air conditioner settings or shutting of the vehicle?
Just from your description it maybe the temperature door inside the A/C box in the rear of the vehicle. A small motor controlled by the A/C computer moves a door directing cold air or heated air. Usually, what I see is this motor fails and stops in either the cold side or hot side. It is also possible the door inside has broke causing only heated air to flow.