No the question is, can it be cheesed by having someone already at floor 120 before starting the quest. I've never tried, so I'd definitely recommend giving it a whirl and seeing what happens. It may work, it may not, they may be kicked out of the mines, the game may crash, who knows. But it's data worth gathering anyway imho.
Personally my hypothesis is that it wouldn't work because the correct "version" of floor 120 (and the quest completion trigger) won't be that which they are in. As such the quest will be picked up and nothing else will happen until they leave the mines, at which point you're stuck. The plan which imho has a greater chance of success is for them to be on floor 119, have found an exit but not gone down it, and Then start the quest. Then, they go down the ladder to floor 120, and maybe the quest is complete?
Anyway let us know how it goes, the only way to know with certainty is experimentation!