If my Irelia Fervent has Boots of Swiftness and I gank to another battlefield, does that count as me choosing Irelia?

No, using the Ganking ability granted by Boots of Swiftness to move Irelia, Fervent to another battlefield does not count as "choosing" her.

According to Rule 352.10.c, game objects mentioned in the text of an ability are targets (and thus "chosen") unless they are included only as part of a cost, trigger condition, or replacement effect. Furthermore, RiftJudge FAQ #3028 explicitly clarifies that moving a unit (including standard movement) does not count as choosing for the purpose of Irelia's abilities.

Reasoning:

• Ganking is a Passive Ability: As defined in Rule 736.1, Ganking is a passive ability that adds permissions to a unit's movement. It does not involve targeting or choosing the unit itself.
• Movement vs. Choosing: While equipping gear to Irelia counts as choosing her (as established in RiftJudge FAQ #6203 and #7601), the act of moving a unit—even when using a keyword like Ganking—is a distinct game action that does not trigger "choose" effects.

Therefore, moving Irelia via Ganking will not trigger her ability: "When you choose or ready me, give me +1 Might this turn."