Why would a 2008 G6 be shaking at about 45 mph and higher with no snow or ice in tires?

Answer 1

shaking and vibration are some of the more difficult things to diagnose.

Focusing on suspension and steering, assuming it shakes both while giving it gas in gear, and coasting in neutral. (make sure its speed/movement related, not engine/rpm related)

Check your tires for uneven wear or a difference in tire pressure. One of the most common causes of vibration is simply that tires are worn and not worn evenly.

The car is fairly new, but depending on your driving habits, your tires may be worn and need replacement (making sure you rotate your tires every 5000 miles will lengthen their lifespan dramatically and most places will do it for free for life if you buy their tires).

if it isn't tire wear or tire pressure, the next things to look at are your steering joints. Bad CV joints, Tie rods (inner and outer) can cause bumps and shakes that can be hard to pinpoint. You alignment may be off causing these symptoms as well.

Look for torn rubber boots and escaping grease, that's a clear sign of a bad joint that may need some attention.

I know SEARS at least used to do free suspension inspections. That might be a good place to start.