Green’s Dictionary of Slang

not give a fuck v.

also ...a cockeyed fuck, ...a fat fuck, ...a good fuck, ...a monkey’s fuck, ...a motherfuck, ...a ratfuck, ...a red fuck, ...a roaring fuck, ...a royal fuck, ...two fucks, not care a fuck
[note, however, one-time late 18C use in a poem ‘The Discontented Student’; the word is not spelt out, but the rhyme and the context ensure that the absentee can only be ‘fuck’]

[1910s+] (orig. US) to not care at all.