Luxon area exploration as a Kurzick?

Can you explore the Gyala hatchery, Rhea's crater, and Silent turf as a Kurzick? I want my exploration title

Yes, you can. All you have to do is get over there and do the "Befriending the Luxons" quest (obtain 10,000 unspent Luxon Faction).

The game is set up so that you can play both sides on any given character.