If you conquer the Battlefield, Hall of Legends, while playing Reksai Void Burrower, can you trigger Reksai first before the Battlefield's conquer trigger?

Yes — if Rek'Sai is ready when you conquer. Since you control both triggers, you choose the order they go on the Chain (Rule 383.3.d). Put Rek'Sai's "when you conquer" on first; Hall of Legends' "ready your legend" resolves first, then Rek'Sai's ability resolves (exhaust it, reveal 2, play one).

But if Rek'Sai is already exhausted when you conquer (say you already used its conquer ability earlier this turn), its trigger never enters the Chain at all — the exhaust is a cost paid to finalize the trigger, so it can't fire. Hall of Legends readying it afterward comes too late for that same conquer. In that case only Hall of Legends' trigger resolves, and you can only use Rek'Sai's ability on your next conquer.