Green’s Dictionary of Slang

not give a damn v.

also not care a damn, not give/care a black damn, ...a dam, ...a D, ...a darn, ...a dern, ...a domn, ...a doodley damn, ...a drat, ...a durn, ...a good damn, ...a nigger’s damn
[damn n., but note Hind. ??? (d?m) a coin of infinitesimal value (1/25 of a paise), ???? (dam??) a coin of small value (1/4 or 1/8 of a paise)]

[late 18C+] to not care at all.

In phrases

don’t-give-a-damn (adj.) (also don’t-care-a-dam(n), don’t-give-a-shit)

[1960s+] (US) careless, carefree; also as n.; thus don’t-give-a-damnism n.