This quest will trigger automatically as soon as you leave Fort Hagen after killing Kellogg in the main story quest "Reunions". You'll see the Brotherhood of Steel's airship fly overhead, eventually mooring in the Boston Airport.
Tune in to the military frequency to hear about Paladin Danse requesting for reinforcements. Head to the Cambridge Police Station. You’ll find that there are a lot of Brotherhood soldiers guarding the station now and that the interior is redesigned. That said, you can find new loot than it was before. One of the loots is the issue of Guns & Bullets inside the safe in the corner office.
Once inside, talk to Danse then head to the rooftop to board the veritibird. Enjoy the ride until the vertibird docks in the Prydwen. Talk to Lancer Captain Kells for the briefing then head to the command deck to attend the briefing of Elder Maxson.
While talking to Maxson, you’ll get promoted to Knight and be allowed to use the Brotherhood Power Armor. Also, the Boston Airport workshop will now be available for you to occupy and use.
Next, climb to the ship’s deck and talk to Danse. He will suggest touring the ship. Agree to go with him and send your current companion to any of your desired settlements. Danse will also become available as a companion. The quest will be complete and a follow up quest will become available. You’ll also get a misc quest to open your personal container on the ship. You’re also free to take the un-owned items in the ship.