if one of my units become mighty during an enemy turn on a showdown can i use fiora legend ability?

Yes — triggered abilities aren't restricted by whose turn it is. Rule 383.3.c says triggered abilities can be put on the chain during Closed or Open States on any player's turn, so if one of your units becomes Mighty (Might goes from under 5 to 5+) during your opponent's turn — including mid-showdown — Fiora, Grand Duelist's trigger fires.

Since it's a triggered ability rather than an activated one, the "no abilities during Showdown State" default (Rule 343.1.b) and the Main Phase/Open State limit on activated abilities (Rule 145.2) don't apply. You choose whether to pay on finalization, and the cost is exhausting Fiora — so she must be ready (or otherwise able to be exhausted) when the trigger resolves. If she's already exhausted, the trigger still goes on the chain but you can't pay, so nothing happens.

Note the trigger only cares about the Might change while the unit is in play, not about the phase or turn — FAQ 742 is the same situation on the opponent's end step.