In 2151, Hoshi Sato noted that the frequency distortions of the symbiotic lifeform found aboard Enterprise NX-01 looked a lot like the phonetic patterns in certain Andorian dialects. (ENT: "Vox Sola")
In 2154, Ambassador Soval told Shran that his name would be used in the Andorian language as the word for fool in the future just as "Nirak", a Vulcan who failed to stop an invasion, meant the same word in the Vulcan language. (ENT: "Kir'Shara")
In 2287, while in the observation room of the USS Enterprise-A, Sybok explained to Captain James T. Kirk that there were many different names for God in different cultures. As Sybok explained, the name for God in the Andorian language was apparently unpronounceable. (Star Trek V: The Final Frontier)
According to the non-canon novel Worlds of Star Trek: Deep Space Nine, Volume 1, the primary spoken language of the Andorians was called "Andorii." An Andorian language glossary is available at the back of that novel. The current novel-universe also uses a different naming convention for Andorian characters.
The Star Trek RPG supplement Among the Clans from Last Unicorn Games, on the other hand, claims the primary language was called "Graalek"; this looks very different from the written language seen on screen, consisting of curved horizontal lines and circles, though this can be explained as two different languages used on the planet. That book attempts to explain the Andor/Andoria inconsistency by claiming that in Andorian, "Andor" merely means "World", but adding "-IA" (Andoria) promotes a word to prominence over other words.