We won't know if they do right now for another several decades. But even if they do, there are good arguments against them being able to break AES - AES is used by the government in contexts that require the information be kept secure for decades, which they could not do if they reasonably suspected AES would be broken in the near future and certainly not if they already knew.

The Enigma case doesn't really apply here - the US was not using the known flawed encryption themselves, and tipping the public off by calling for the development of a new standard only means that AES might be broken in the next decade or two, and maybe not by quantum computers. Cryptographers already recommend switching from weak algorithms years before they're fully broken.

RSA and other key exchange techniques are another story though, as the article mentions RSA is theoretically significantly weaker to a functioning quantum computer than AES to start with, and I don't really know the extent to which the government uses RSA.