Sadly, we don't know.
All Tolkien says of dragon-language is a brief note during a discussion of the spread of Westron (emphasis mine):
In Bilbo's time there was a language very widely used all over the West (the Western parts of the Great Lands in those days). It was a sort of lingua-franca, made up of all sorts of languages, but the Elvish language (of the North West) for the most part. It was called the Western language or Common Speech; and in Bilbo's time had already passed eastward over the Misty Mountains and reached Lake Town, and Beorn, and even Smaug (dragons being ready linguists in all ages).
History of Middle-earth XII The Peoples of Middle-earth Chapter 2: "The Appendix on Languages"
Bearing out Tolkien's point, on the rare occasions we see dragons speaking they never have trouble being understood. So they're evidently capable of at least speaking Sindarin and Westron, and presumably could learn most other languages if it came down to it.
A smart question would be to ask what language dragons speak amongst each other, but unfortunately we don't know that either: the occasions in the book where we see more than one dragon at a time are rare, and exclusively battles1. Hardly an environment conducive to conversation.
There is evidence that Morgoth created the Orcish tongue in the First Age; a draft of what would eventually become Appendix F, dated to no later than July 1950, contains the following paragraph:
The Orcs had a language of their own, devised for them by the Dark Lord of old, but it was so full of harsh and hideous sounds and vile words that other mouths found it difficult to compass, and few indeed were willing to make the attempt.
History of Middle-earth XII The Peoples of Middle-earth Chapter 2: "The Appendix on Languages"
If he did this for the Orcs, it's possible that he would have done the same with Dragons, or taught the Dragons the Orc-speech, but we have no evidence to support either theory.
1 In fact I think there might be only one example, the Fall of Gondolin. But it's possible I'm forgetting some