Now that you have dealt with the matter of the inquisitor, you seek an audience with Lord Drillemont to receive formal permission to enter the Stone Vigil.
This quest is available if you have completed the quest The Heretic Among Us, and have reached level 41.
Starting the Quest
Talk to Alphinaud in the Whitebrim Front in Coerthas Central Highlands to begin the quest.
Speak with Drillemont
Go south from the fire pit and enter the building, and go up the stairs. Talk to Drillemont and there will be a cutscene.
Speak with Nathelain
Go downstairs and go outside, then go through the northwest archway. Make your way northwest to where the orange arrow is pointing in the minimap. Talk to Nathelain to add the Stone Vigil to your Duty Finder.
The Stone Vigil
Use the Duty Finder or Duty Support to enter The Stone Vigil. Please see the The Stone Vigil Guide for a walkthrough.
After you defeat the dungeon boss and exit the dungeon, there will be a cutscene.
Speak with Alphinaud
After you complete the Stone Vigil, talk to Alphinaud to complete the quest.
Next Main Scenario Quest
After you complete the quest, talk to Cid to accept the next quest: Into the Eye of the Storm. There will be a cutscene.