Green’s Dictionary of Slang

dog n.4

[? the animal’s inability to speak]

(US) nothing; usu. in phr. never say dog, to stay silent.

[US] in N.E. Eliason Tarheel Talk (1956) 154: At 8 the ladies retired & said not dog.
[US]Green-Mountain Freeman (Montpelier, VT) 15 Aug. 4/1: Never say ‘dog’ to me [...] and I’ll treat you to a dinner!
[US]DN I 376: Where’s that boy? He went off and never said dog.
G.P. Smith ‘Speech Currents in “Egypt”’ in AS XVII:3 170: I never said dog to them about it.