Unlocking world quests in Nazjatar is a part of the questline there involving the NPCs who become your bodyguards.
The actual unlock itself is partway through the 2nd chapter of the zone, Secrets in the Sea.
To get there though you must have completed the first chapter, Welcome to Nazjatar. You’ll notice in the guide here that Welcome to Nazjatar has a part involving Magni and the Heart of Azeroth.
The Magni questline (which starts here) must be completed to move onto the next stage of the Nazjatar storyline.
If you aren’t seeing the world quests in the zone, you aren’t far enough along in the questline to have unlocked them yet.