Does Fluorescent Lighting Cause GFCI-Protected Circuits to Trip?
Most likely, it's an actual ground-fault in a defective ballast, especially in light of the fact that one fixture is causing it to trip.  In the past couple decades there's been a switch to electronic ballasts, starting with the higher-end units.  I see LOTS of failed, older magnetic ballasts and a few failed cheapie electronic ballasts.  

A quality fluorescent fixture made out of stout sheetmetal is worth keeping, because it's infinitely upgradeable to better ballasts, T8, even LED. 



So I would first, check the lampholders for any signs of physical damage or arcing.  Tombstone style ones cost about a buck and they're mostly standard. 

Then I'd swap the ballas-- hold on, I'd pause and evaluate LED replacement "tubes" -- particularly for build quality, light quality and color.  There is an ocean of cheap LED "tubes" out there but the quality isn't very good.  To be honest I'm sticking with fluorescent because I can walk into Menards and get a top-name 90 CRI bulb in either T8 or T12 for a sane price. You won't see a 90 CRI LED at a consumerable price, for awhile yet.   I'll wait. 

Since I'm doing the ballast, I'd make a call whether to stay with obsolete T12 tubes or switch to the superior T8.  It matters - the ballast must match the bulbs.  I've seen T8 tubes in a T12 ballast or vice versa, but it runs poorly and fails the bulbs quickly.  

In the USA, we "outlaw" old tech by requiring quality comparable to the new tech.  If the market can do it, good on them.  So T12 bulbs are still for sale, and they have excellent efficiency and CRI.  But I wouldn't count on that long-term.  

Lastly I'd evaluate instant start, vs rapid-start, vs programmed-start ballasts.  Instant-start wears out the tubes fast.  Rapid-start uses the pre-heat filaments (hence, 2 wires to each end of the tube) to warm-start the light.  Programmed-start ballasts use a microcomputer to do that very well. That matters if the bulb is hard-to-reach, expensive or switched on often. 

As you might guess, instant-start ballasts uses 1 wire per tube.  They are often used with shorting lampholders, which short the two pins together. Those won't play nice (or even connect!) to a rapid- or programmed- start ballast, you'd need to change lampholders. 

Lastly, your ground fault may come from a tendency in some ballasts to use capacitive effects between bulb and metal frame of the fixture to start the bulb.  These are cheaper ballasts, and yes, that thing they do is an intentional ground fault, which would trip a GFCI unless it's isolated (i.e. by a transformer). 



Noting that you linked a $13 Utilitech fixture as your likely fixture, as you can guess, those are pieces o'junk.  I regrettably have about a half dozen.  I've had 2-3 ballast failures in 6 years! They usually use instant-start ballasts, and judge the sheetmetal for yourself.  The lampholders are alright... but they're shorting.   Cord's good.  All 8 inches of it.