Wedge a guitar pick or toothpick on either side of the saddle. Not the part bolted to the baseplate, but the part that the string goes into. If that stops it, you have a loose saddle. That means the area where the pin goes through the saddle, and therefore holds it to the other part in front of it, has too much play in it. They have to have some play for the fine tuners to work but sometimes the saddle will rattle, or the pin rattles. (Dr Seuss look out) If so, you can put a blob of linseed oil in there, where the pin goes through, and let it dry. It won't actually dry, it'll just get gummy, and should stop the noise.
But it really could be anything, while the G string just happens to be at the frequency or the location on the bridge to bring it out.