What should I do about no ground wire
The danger of connecting the green wire to a ground that doesn't go all the way back to the panel is that if the fan has a ground fault, the ground wire will not carry the fault back to the safety of the panel, and instead will distribute the ground fault to everywhere on that island of grounds.  So all the grounds will suddenly become hot. 

I would leave it disconnected.

I would also assure that this section of the circuit is protected by GFCI.  

I would also see that the entire circuit is protected by an AFCI breaker back at the panel.  

These have totally different jobs; the AFCI protects from arcing overheat loads caused by failing wires.  The GFCI protects humans from shock due to lack of grounding.