I have been asked to install a GFCI breaker on a lighting cir. With just 3 flouresent lights but the breaker won't hold. Will flouresent lights work with GFCI breakers?

Lighting is not required to have GFI protection, and it is not recommended, exactly for the reason as you can see by the problem your having.

Ballasts usually have minor fault currents, and do not work well with GFI.

Who wants this and why?

I suspect you mean AFCI.
I also ask, who is telling you this and why?