Following the Hunter's attacks on the Lodge and the Eye at the end of High Price to Pay, Matteo will suggest talking to Keeper Aquilus at the Sanctum Universum. Doing so triggers the Unity main story mission, which has players traveling all over the universe to find information about the eponymous Unity and a mysterious traveler known only as the Pilgrim.
Starfield: How to Open the Pilgrim’s Room (Pilgrim's Rest Quiz Answers)
Their quest to understand the Unity will lead Starfield players to Pilgrim's Rest, where they'll need to complete a quiz to unlock the Pilgrim's room.