Speaking with Elder Maxson again after having met everyone of note within the Prydwen, the elder will then proceed to prep the Sole Survivor on their next mission, in which it is revealed that the nearby Fort Strong is sitting on an unusually large cache of mini nukes that the BoS will need in their war efforts against the Institute and their monstrosities.
After being briefed on the mission, the Sole Survivor will need to return to the flight deck and enter a prepped vertibird, to be transported to the outskirts of Fort Strong, where the Sole Survivor will need to use the vertibird's minigun to take down a group of super mutants, finally cultivating in an encounter with a super mutant behemoth.
After taking down the behemoth and entering the ruins of Fort Strong, it is discovered that super mutants have completely taken over the area, and must be routed out in order to gain access to the fort's lower levels.
After clearing out the super mutants, and gaining access to the fort's armory and its impressive stores of mini nuke parts (which are seen to need further assembly), speak to Paladin Danse, before returning to the Prydwen via vertibird or fast-travel.
Meet with Elder Maxson
Now that I've familiarized myself with the Prydwen and her crew, I'm supposed to meet Elder Maxson on the Flight Deck for my first mission.
Board the Vertibird Gunship
Elder Maxson has ordered me to head out to nearby Fort Strong and cleanse it of all hostile Super Mutant forces. There's a vertibird standing by on the Prydwen's Flight Deck that will fly me to the location.
Secure Fort Strong Armory
Secure Fort Strong
Kill the Super Mutant Behemoth
Speak to Paladin Danse
I've exterminated all Super Mutant forces at Fort Strong. I should link speak to Paladin Danse before I head back to the Prydwen.
Speak to Elder Maxson
Paladin Danse has ordered me to return to the Prydwen to speak with Elder Maxson.
The vertibird's gun has unlimited ammunition, so use it to eliminate all hostiles in the area, not just the behemoth.
If Danse is the player's companion, he will leave the player's service and return to the Prydwen when the armory is secured.
After completing this quest, Elder Maxson will reward the Sole Survivor with the ability to call in BoS vertibirds for transportation purposes. Due to limitations on vertibird numbers, they will not stick around long for combat roles, and in order to call them in, rather pricey signal grenades must be bought from Proctor Teagan.
If the vertibird cannot be entered, and the quest's objectives refuse to update, try entering the Prydwen and coming back out.