Can I add a green ground screen to my exhaust fan's housing (metal) and connect the green ground wire to it? There is no ground wire in the old supply wiring.
Sure, but it won't do anything. The term "ground" implies an electrical connection to the Earth, which is normally provided by the bare grounded conductor.
If the "old supply wiring" has no ground, and ground is not available by conduit or nearby plumbing, then you would have to install a new cable that contains a ground wire. Unfortunately, the NEC does not permit using the neutral (aka grounded conductor) as the equipment ground per 250.136(B)