Fluorescent light trips adjacent GFCI
Likelihood 1: Since the GFCI is the new thing, it is miswired.

This usually happens when someone is replacing a plain receptacle with the GFCI, hooks up the supply wires to the two LINE screws, and finds themselves with wires left over.   Those used to go on the extra screws on the receptacle, which were for the purpose of being spares.  However, GFCIs don't have any spares, just the two LINE screws.  

This is where novices typically peek under the warning tape that says "Do Not Use, for wizards only" in as many words, and they see two more screws. Voila, somewhere to attach my spare wires!   Except those aren't spares, they are the LOAD terminals intended for extending GFCI protection to other outlets.

Sometimes this mistake inadvertently leads to extending GFCI protection to other circuits, yay I guess.  Other times, this is miswired, and causes nuisance trips.   

Likelihood 2: Or your light has a ground fault

Suppose you build an X Detector because X is a common failure point in your situation.  And it immediately detects X.  The detector must be bad, right?  That is exactly how people react to GFCI trips. One guy bought $75 of new GFCIs because he could not believe his $30 coffeemaker had a ground fault.  

I imagine that you did not have a GFCI before.   So maybe the lamp has had a ground fault this entire time, and you just didn't know it.  You can test that by removing the ground wire temporarily, so the fixture has no ground; but beware: its chassis will be energized at 120V if it in fact has a ground fault!  Don't leave it that way, no matter how much it seems like an easy fix. 

That's not a surprise with old fluorescent magnetic ballasts. I would say in your light upgrade, you "zigged when you should have zagged".  Buying expensive LEDs that absolutely depend on maintaining a ballast does not protect you from failure of the ballast. Instead I would have bought a quality electronic ballast cheaper than the LED "tubes", and just stayed with real fluorescent.   Now you have two choices: 


replace the ballast  (and retain the ability to rollback to real fluorescent if those LEDs turn out to be a piece o' junk) -- consult their instructions for the type of ballast to buy; it matters. 
sell those LED "tubes" on craigslist and buy direct-wire LED "tubes", and rewire the fixture to bypass the ballast and feed 120V straight to the LEDs.