Can I play the effect of Gust Monk from my enemy trash?

Yes — Gust Monk's trigger reads "banish a card from any trash," and "any trash" includes your opponent's. So when you play Gust Monk and pay the [1] additional cost, you can banish a card out of the enemy trash to give a unit Assault 2 this turn (FAQ #11937).

One clarification on the wording: the effect isn't something you play from a trash — it's a "When you play me" trigger on the Gust Monk you played, and the enemy trash is just where you take the banished card from. The banish is a cost-within-instruction, so it has to be paid when the trigger is placed on the Chain; if both trashes are empty, the trigger can't finalize and fizzles, though Gust Monk still enters play (FAQ #11936).

And you can't play Gust Monk itself out of your opponent's trash — a card in a trash is only playable if an effect grants that permission, and their trash isn't your zone to play from.