Before Smaug awakens because of Bilbo's ventures, there's not even a single breath amidst all the gold. His breath is only heard when he wakes up. Is he capable of living without breathing?
In order of possibility:
Smaug does not need to breathe. He is a magical monster and his life processes do not resemble that of living beings. After all, he can breathe fire, fly AND speak intelligently. Of the magical origins of dragons little is actually known so they may not have the same requirements as non-magical creatures. Dragons violate so many of the natural laws of physics as to be certainly supernatural in and of themselves.
Smaug breathes, but while dormant under his gold may only breathe once in a given period which was shorter than Bilbo's initial visit. Like many serpents and reptiles, they are able to live for very long periods without any environmental resources. For example: The bushmaster viper may live an entire year on less than one meal per month.
Smaug breathes very quietly unless he wants to talk, interact, or breath fire. He may when in a hibernation state, need almost no natural resources, such as air, at all.
It has been suggested that possibly like an alligator, Smaug may lie below the gold for the most part but his nostrils may be camouflaged by the gold, and the gold may be loose enough to allow him to breathe quietly while at rest.