06 G6 power Steering kicking in on right turns

Not all power steering systems are hydraulic. I think some G6 models have electronic power steering so no fluid is needed. Referring to the owners manual it appears that only the 3.9L uses a hydraulic system. If there is a reservoir it will either be tucked behind the alternator or mounted on that end of the engine. The top mounted reservoir filler cap is on top of the engine to the left of the oil filler cap.

Maybe it doesn't have a place to put power steering fluid ! Could be it has Electronic Power Steering (electric )
Power Steering Motor
The power steering motor is a 12-volt brushless DC reversible motor with a 65-amp rating. The motor assists steering through a worm gear and reduction gear located in the steering column housing.

Bad battery? If you are positive the battery is ok, maybe something is staying on and draining it. parasitic draw test YouTube

power steering pump valve sticking .

it will get no avail here either unless yo provide some sort of detail.... front, back, left right?? steering, suspension, braking?

In 2006 Pontiac put ELECTRIC power steering in the G6's.

Your looking under the hood in the right location its just under that. its all electrical. Must have been not such a good idea because they switched back to normal power steering in 2007.

(I have an 06 G6 V6)

known issue... there are thousands of people with the same problem. It's the same issue in the G5 (recall issued for G5, but not G6) issue is that your electric steering has some loose wire connectors on the shaft.

in plain sight in the engine compartment.... unless you have the infamous electric power steering.

some G6 do not have power streeing pumps
they have ELECTRIC POWER STEERING
all depends on the engine
2.2 4cylinder have electric
3.5V6 can have one or the other
if you have a V6 then you will see the power steering cap through the engine cover on the right side if it has hydralic power steering

I suggest that you verify fluid quantity and belt tension.