This quest is not available in game.
Go to Stormwind's Old Town and either speak to a Doomsayer or take a robe from the crate near a Doomsayer to join the doomsday cult, then equip the Doomsday Recruit's Robe you receive.
It's getting difficult to keep the cultists out of the public eye and rumors are spreading across the city. I want you to join the cult and learn as much as you can.
Approach one of the doomsayers in Old Town and tell them you wish to join their group. The doomsayer will likely give you clothing and instructions to meet with other cultists in one of the group's hideouts. Follow those instructions, <name>. It's the only way to gain their confidence.