Actually it sounds like a wiring issue. Conductors have an inductive/
capacitive link between them. If you have a stray conductor running a
different load that is close to and parallel to the wires feeding the
fan motor, then you'll get a current imbalance through the GFCI.

Nope, other than where the cable exits the switch box, there are no other
cables anywhere near the cable running to the ceiling fan.

Also, the fan NEVER trips while running, only when it is turned OFF. And
even then, it's only once every few weeks or so.

Anthony