flash choose irelia?

Yes — playing Flash and choosing Irelia, Fervent triggers her ability, giving her +1 Might this turn.

Flash's "move up to 2 friendly units to base" chooses those units, and her trigger is checked at finalization, when the spell is placed on the chain — so the +1 goes on the chain above Flash and resolves before the move happens. It's still there after she lands at Base; Might bonuses don't care about location, and the move doesn't ready her, so you don't get a second +1 from the "ready" half.

Note her Deflect only taxes opponents choosing her, so it does nothing when you target her with your own Flash. This is standard "when chosen" behavior (FAQ #9163).