Can I play gust monk and have no cards in the discard can I target my irelia fervent with assault 2 even though I have no cards in discard

No — if there are no cards in any trash, you can't give Irelia, Fervent Assault 2.

Gust Monk's trigger requires choosing "a card from any trash" as part of its effect. When the trigger finalizes, if there are no legal options for that choice, the ability is removed from the chain and never resolves (Rule 402.4). It also reads "banish a card from any trash to give a unit Assault 2," so the buff is dependent on the banish actually happening — same principle as "discard to" wording (FAQ #4621).

Note "any trash" means either player's. If even one card exists in your or your opponent's trash, you can banish it and give Irelia Assault 2; Gust Monk itself still enters play fine either way — only the trigger fizzles.