Green’s Dictionary of Slang

(orig. US) to philander.

1939		J. Steinbeck Grapes of Wrath (1951) 76: He’s a billy-goatin’ aroun’ the country. Tom-cattin’ hisself to death.
1967		J.E. Schmidt Lex. of Sex 20: Billy-goat [...] Of a male, to engage in extra-cardiac love affairs, via the pudendal route [HDAS].