New GFCI in garage is tripping when driveway lights are turned on
Sounds pretty clear to me (and likely) that there was a phase to ground fault in the lights before you added the GFCI, that went unnoticed until now.

They might be wired wrong (if someone swapped ground and neutral).  But most likely, there's a fault connecting neutral and ground somewhere beyond the switch.

To confirm this, disconnect the lights from the switch and use a multimeter to measure resistance between hot and neutral, then hot to ground and neutral to ground.  If things were in good working order, you would have infinite resistance (no connection) between ground and neutral, and between ground and hot.  Instead, you will likely find one of the two is conducting to ground.  To locate this fault, you will have to divide the system at each wiring junction and re-test.

One last possibility (given this particular circuit is outdoors) is that the ground wire to the lights is electrically isolated from the physical soil, and that current is leaking from hot into the soil.  This would be more difficult to test.

I would narrow in on any outdoor construction that took place or any spikes put into the earth near where the wire is buried.  Probably something nicked the insulation and the moisture in the earth is leaking current from it.