Not much is said explicitly about Smaug's age in Tolkien's writings. So pinpointing his age with precision is going to be impossible. However, we can draw some inferences about his approximate age.
I have seen, several times, the suggestion that if Smaug is really one of the most powerful of the winged firedrakes, then he may actually have been birthed in the First Age, one of the first cohort of flying dragons that were released by Morgoth at the climax of the War of Wrath. However, based on some remarks by Gandalf, I think this interpretation is untenable, and Smaug must actually be much younger than that.
The key quotes come from this section of "An Unexpected Party," as the characters are discussing the side door:
“It may have been secret once,” said Thorin, “but how do we know that it is secret any longer? Old Smaug has lived there long enough now to find out anything there is to know about those caves.”
“He may — but he can’t have used it for years and years.”
“Because it is too small. ‘Five feet high the door and three may walk abreast’ say the runes, but Smaug could not creep into a hole that size, not even when he was a young dragon, certainly not after devouring so many of the dwarves and men of Dale.”
It is clear that Gandalf thinks that Smaug might have been able to fit through the opening when he first conquered Erebor, at which time he was "young."
The inference that he was fairly young (not necessarily even fully grown) is further reinforced by the fact that Smaug came to the Lonely Mountain from the North. According to Thror's Map: "Far to the North are the Gray Mountains & the Withered Heath whence came the Great Worms." Moreover, Thorin describes Withered Heath as "where the great dragons bred." So it seems likely that Smaug had actually hatched there, in the wastes of the North, and his assault on Erebor may have been his first foray beyond the Withered Heath's environs.
There remains the final question of how old "young" really means for a dragon. There is some guidance provided by Smaug's own words. In his famous oration to Bilbo, the dragon boasts:
I laid low the warriors of old and their like is not in the world today. Then I was but young and tender. Now I am old and strong, strong, strong, Thief in the Shadows!
If Smaug was young when he captured the Lonely Mountain, enough time has passed to make him "old." Smaug's dominion over Erebor lasts 171 years, from T. A. 2770 to his death in T. A. 2941.
A time to maturity of (very roughly) one hundred years also accords with what is known of draconic biology from The Silmarillion; over the six-hundred-year course of the War of the Jewels, Morgoth was able to breed multiple generations of dragons, improving from the lizard-like Glaurung to the mammoth, flying Ancalagon. Moreover, while Glaurung's date of hatching is not recorded, he first appeared as a weapon in during the Siege of Angband, in F. A. 260, having not been present at the Dagor Aglareb in F. A. 60. So most likely, the Father of Dragons had grown from infancy to adulthood during those two hundred years, while the Noldor maintained there leager around the Dark Lord's citadel.
In summary, we know that Smaug, when he arrived to attack Erebor was "young," possibly less than a hundred years old, and probably not actually fully grown. After 171 years as usurper King Under the Mountain, he considered himself old, possibly somewhere in the middle of his third century of life.