The first thing I'd check is the tuner itself to be sure it's installed and operating correctly. I honestly can't imagine a slack tuner actually tightening a string (causing it to become sharp), but if the "G" string post is not perpendicular to the headstock like the other five posts, it could be that as the neck naturally flexes with changes in temperature & humidity, and as the other strings become slightly sharp, a cockeyed tuner post on the "G" string makes it much more sharp. The fix could be as simple as loosening, reseating, and re-tightening the tuner to the headstock.
Not wanting to sound like the voice of doom-n-gloom, but about the only other thing I can think of to make just that one string go consistently sharp is that the neck or headstock is twisting -- i.e., warping unevenly along the long axis of the neck. As good as Taylor's quality control is, sometimes a wonky piece of wood will slip through and become a problematic neck.
In either case, you may not be able to remedy the problem yourself. A trip back to the Taylor dealer who sold you the guitar would be the first thing. Explain the problem you're experiencing to them. They may be able to correct it for you in the store. If not, the guitar will be sent either to a Taylor service center or back to the factory for warranty repairs. Shouldn't cost you anything if you're the original owner of the guitar. Ultimately, rest assured that Taylor will make it right.
