Green’s Dictionary of Slang

to talk, to chat with.

1958		F. Norman in Sun. Graphic 23 Nov. in Norman’s London (1969) 39: I saw one of the very biggest villains in the entire world bunnying away [...] to a double cultured geezer with grey barnet.
1989		J. Morton Lowspeak 32: Bunny – to talk indiscriminately.