Is it a single click or a rapid clicking? A single click may mean the starter motor has failed. Rapid clicking can be related to the starter solenoid. I think they are a one-piece unit on the G6. One thing I would try if it were my car is taping on the starter solenoid or starter with a rubber mallet or similar. The tapping might free the solenoid or jiggle the starter motor so the brushes to the motor make better temporary contact. If you find you are tapping it regularly to get it going, it is a sign that it will need to be replaced. Good luck getting it going again
I'm pretty sure you can test if it's the starter by jumping pins 87 and 30 of the starter relay. If the starter spins that would point to a bad relay. Or try removing the starter relay #31 and using #28 or #30 in it's place and see if that works. Before trying any of these suggestions search YouTube for how to jump a relay, if done incorrectly you can fry the cars computer.