Green’s Dictionary of Slang

1. (orig. US) to brag, to boast.

1941		(con. 1938) A. Lomax Mister Jelly Roll (1952) 235: ‘Handy could afford to be charitable and slightly contemptuous of Morton, the old whorehouse pianist who seemed to be trying to loud-mouth his way back to big time [...]’ one critic observed.
1961		C. Himes Pinktoes (1989) 24: More singing and dancing, more knife-toting and loud-mouthing.

2. (US black) to speak abusively; as a synon. term for dozens n. (1)

1938		Charles ‘Cow Cow’ Davenport [song title] Don’t You Loud Mouth Me.
1964		Muhammed Ali in Playboy Oct. 🌐 Well, the big thing I did is that until then, I had just been loudmouthing mostly for the public to hear me, to build up gates for my fights.
1974		H.L. Foster Playin’ the Dozens 198: Sometimes, ‘loud mouthing’ or ‘loud talking,’ ‘sounding,’ ‘screaming on someone,’ or even ‘bogarding’ are synonyms for ‘woofing’.
1988		W.D. Myers Scorpions 18: ‘I went down there [...] and he came loudmouthing me in front of everybody.’.