Given the opposing ideologies of The Institute and, well, everyone else, it shouldn't be surprising that some non-Institute quests are triggered by alienating yourself from Father and his followers. Quests like Spoils of War and Form Ranks can be triggered during this event.
There are a couple methods to completing this quest. The most civil route would be to disagree with everything Father and his division leaders say. The more dramatic way would be to simply kill Father. If you prefer the latter, you should do one thing before committing infanticide: Call the elevator in the Institute concourse — but don't ride it just yet.
Meet Father in his living quarter and kill him.
You'll never witness another faction so focused on one goal: your demise. The game will prevent you from fast travelling, so your only means of escape will be to use the mass relay on the top floor. This is why you should have the concourse elevator at the ready. Ride it up to the relay and use the teleporter to escape.
You'll appear at the ruins of C.I.T., closed off from all future Institute missions. If you happen to pursue this quest with the Underground Undercover quest active, you will also close yourself off from future missions with The Railroad.