Green’s Dictionary of Slang

(orig. US) a euph. for a damn n.; thus I don’t care a monkey’s fuck, I don’t give a monkey’s ass etc.

1960		G.W. Target Teachers 100: The Old Man’s door opened and the pair of them came out, Stillwell not seeming to give a monkey’s, but too casual, and poor Jimmy Taylor with his hands clenched before him like the broken forelegs of a ginned rabbit .
1962		B. McGhee Cut and Run (1963) 44: I felt like telling him to go and take a ‘monkey’s’.
1970		G.F. Newman Sir, You Bastard 69: Your old man don’t give a monkey’s.
1999		J. Cameron Brown Bread in Wengen [ebook] Never gave a monkey’s for my hard luck story.
2009		J. Joso Soothing Music for Stray Cats 93: ‘I don’t think that’s for you to say.’ I says, ‘I don’t give a monkey’s pal.’.
2010		K. Barry ‘Fjord of Killary’ in New Yorker 24 Jan. 🌐 ‘I’ll bury anythin’ that fuckin’ moves,” he said. [...] ‘I couldn’t give a fuckin’ monkey’s,’ he said.
2011		Times Mag. 30 Apr. 14/1: I don’t give a monkey’s.