This question is used when we want to express an illusory interest in a given topic.
Majkela: Filip, I'm googling how to give a fuck about what you talking about.
Filip: Ok
Unfortunately, this department cannot assist you with your problem. Why? Because I don't give a fuck.
Joey just got fired. He wants to appeal it.
He needs to talk to the department of I don't give a fuck. That guy's an idiot.