If I buff Irelia Fervent with Blind Monk, does she get +1 from Choose?

Yes, Irelia, Fervent will gain +1 Might from her ability when you use Blind Monk's ability to buff her.

According to RiftJudge FAQ #9143 and #9507, the act of choosing a unit to buff it counts as "choosing" or "targeting" that unit. Since Irelia, Fervent's ability states, "When you choose or ready me, give me +1 Might this turn," the act of choosing her as the target for Blind Monk's ability triggers her bonus.