Green’s Dictionary of Slang

(US) to brag (about); thus bigmouthing n., bragging.

1953		Kramer & Karr Teen-Age Gangs 24: I just get tired of big-mouthing. I hate a guy to take a penny and turn it into a fifty-dollar bill.
1957		C. Himes Imabelle 40: He was big-mouthing to my girls about how he was going to make a fortune.
1965		C. Himes Rage in Harlem (1969) 42: [as 1957].
1968		‘Hy Lit’ Hy Lit’s Unbelievable Dict. of Hip Words 39: talkin’ trash – Telling a lot of lies, exaggerating [...] big mouthin’.