Green’s Dictionary of Slang

(US) the desire to do something, the state of being motivated towards, enthusiastic for.

1976		oral testimony in Lighter HDAS II 899: That’s not a question of talent; it’s a question of give-a-shit.
1985		(con. 1968) D.A. Dye Citadel (1989) 245: I can’t seem to work up a good give-a-shit about it.
2006		G. Pelecanos Night Gardener 65: [...] trying to put a tone of give-a-shit in his voice.