When Rek'Sai Swarm Queen attempts to conquer an empty battlefield, does her passive trigger to play a unit there?

No, Rek'Sai, Swarm Queen's ability does not trigger when you move her to an unoccupied battlefield.

According to the established rules and RiftJudge FAQ #8827:

1. Open Showdown: When you move Rek'Sai to an empty battlefield, you initiate an "open showdown." At this stage, there is no combat, and no units have the "Attacker" or "Defender" designation.
2. Trigger Condition: Rek'Sai's ability ("When I attack...") is an Attack Trigger. According to Rule 383.4.e, these triggers only occur when a unit gains the Attacker designation during a combat showdown.
3. Result: Because moving to an empty battlefield does not initiate combat, Rek'Sai does not gain the "Attacker" designation, and therefore her ability does not trigger.

If an opponent later moves a unit into that same battlefield, the open showdown ends and a combat showdown begins. At that moment, Rek'Sai gains the "Attacker" designation, and her ability will trigger then.