MINUTEMEN QUEST 09 – With Our Powers Combined
Triggered by becoming hostile with the Brotherhood of Steel after completing The Nuclear Option for any non-Institute faction. This can also be started after completing the main questline for a faction.
Note: Firing on the Prydwen later in this quest will trigger Minutemen Quest 07 – Defend The Castle, however the Institute forces will be replaced by Brotherhood of Steel troops arriving by vertibirds. This is a long fight with tough opponents, make sure you are well prepared and the Castle is well defended before you trigger this fight. Yes, you do the same quest twice (it appears twice in the questlog) if you proceed with this quest.
Objective: Talk to Preston Garvey
Once you are hostile with the Brotherhood of Steel and have finished The Nuclear Option for any faction (including the Brotherhood) he will inform you that it's time to bring the might of the Minutemen to bear against the Brotherhood of Steel. To do this, he want you to build artillery placements in multiple settlements to fire upon the Prydwen from several directions, preventing the Brotherhood from defending against the attack.
Objective: Build artillery in 5 settlements
You heard the man, build, build, build! Hopefully you've got a decent stash of junk at this point, otherwise you'll be taking a while to collect the oil, springs and other components to build 5 of these damn things. If you already build one in the Castle for the earlier quest Old Guns and manned it, you'll thankfully only need 4 more. Go to four settlements WITH SETTLERS (as they need to be assigned to make the artillery count towards the total) to build them.
Objective: Launch artillery strike on the Prydwen
Travel to the Castle and talk to the radio operator in the center of the courtyard to launch the strike.
Objective: Destroy the Prydwen
Automatic. Watch it fall from the sky (at least if you had a clearer day than my playthrough).
Objective: Defeat Brotherhood attack on the Castle
Minutemen Quest 07 – Defend the Castle will trigger here with the same objectives and percentage indicator as the fight progresses. The only change is the enemies you fight. Nearly every enemy will be wearing power armour and use laser weapons, so you'll want a good supply of stimpaks and to use cover effectively while you pick away at them. For those with plenty of ammo to spare, you can use missiles to bring down the vertibirds and their passengers extremely quickly. Even regular gunfire from a powerful gun will do it and save you killing the soldiers on board.
Objective: Talk to Preston Garvey
When the incoming attack has been crushed, talk to Preston to finish things up. Looks like things are done once and for all as the remnants of the Brotherhood scatter to the winds.