I seem to remember reading in the gazeteer that they did, but I could be wrong. It would be cool if they had a variant of Taldane... They do seem to be the French to Taldors English... Let alone Absaloms Brittain...
I thought Kelesh was the language of the Keleshites.
It doesn't say that Chelaxians have their own language in the campaign setting, nor the Taldans. What is the other source that says they gain a language besides just common?
References to "Chelaxian" as a language are errors.
The base language spoken by Chelish citizens, be they full-blooded Chelaxians or otherwise, is Common (AKA: Taldane). The high mucky-mucks in the Chelish government may at times speak Infernal, though.