Surprisingly, many of these exist. To list a few there's the Aztec belief:
The Aztecs' story maintains that only a man, Coxcox, and a woman, Xochiquetzal, survive, having floated on a piece of bark. They found themselves on land and begot many children who were at first born unable to speak, but subsequently, upon the arrival of a dove were endowed with language, although each one was given a different speech such that they could not understand one another.
There's the greek myth about linguistic diversity, which I find to be interesting, although not very unique:
In Ancient Greece there was a myth which told that for ages men had lived without law under the rule of Zeus and speaking one language, gifted to them by the god and goddess of ingenuity, Philarios and Philarion. The god Hermes brought diversity in speech and along with it separation into nations and discord ensued.
One that I find interesting in particular is a south Australian myth:
In remote time an old woman, named Wurruri lived towards the east and generally walked with a large stick in her hand, to scatter the fires around which others were sleeping, Wurruri at length died. Greatly delighted at this circumstance, they sent messengers in all directions to give notice of her death; men, women and children came, not to lament, but to show their joy. The Raminjerar were the first who fell upon the corpse and began eating the flesh, and immediately began to speak intelligibly. The other tribes to the eastward arriving later, ate the contents of the intestines, which caused them to speak a language slightly different. The northern tribes came last and devoured the intestines and all that remained, and immediately spoke a language differing still more from that of the Raminjerar.
I hope these are helpful and if you want to learn more you can go the Wikipedia page on Mythical origins of language.