Do the countered spells still trigger Irelia, Fervent's Trigger?

Yes, countered spells still trigger Irelia, Fervent's ability.

According to RiftJudge FAQ #636 and #6256, Irelia's ability triggers when she is chosen during the spell's finalization process, which occurs before the opponent has the opportunity to counter the spell. Because the trigger is placed on the chain at the moment of finalization, it remains there and will resolve even if the original spell is subsequently countered.

Sequence of events:

1. Finalization: You play a spell and choose Irelia as the target.
2. Trigger: Irelia's ability triggers and is placed on the chain.
3. Counter: The opponent plays a counter effect (e.g., Defy) targeting your spell.
4. Resolution: Even though the spell is countered and sent to the trash, Irelia's triggered ability remains on the chain and resolves, granting her +1 Might.

This principle applies to all "when chosen" or "when targeted" effects in the game.