Endure cooldowns will tick down whenever time advances while your HP and SP are both at maximum: nothing else (including how this happened) matters.
So yes, if Galvanize ends up taking you to full SP and you have full HP, Endure cooldown will begin ticking down.
Note that there are 10 ticks to 1 point of displayed cooldown: if you immediately get hit out of full HP/SP by some means, you may not be at full long enough to actually drop 1 point of displayed cooldown.