Ok, so here is what I know.......like you, I recently bought a 2010 G6 1SH. For the price it's loaded. The only thing it's missing is the Bluetooth. I even stumbled on this forum through your post in my search for information on this subject. There is virtually no information about the 2010's. Most Bluetooth questions come from the pre 2009.5-ers that do not have the capability to use BT (in mid 2009, BT capability was included automatically).
What I found out was that the Fleet/rental cars (which most of these are) where made to order based on cost and the UFP option code (Bluetooth) was not ordered by most companies. Although our cars and radios are new (USB etc.) and BT capable they do not have the Onstar WITH Bluetooth module. Most companies did get the Onstar because these cars are inevitably resold to the public (at least that's what I've read).
Apparently, you can replace the Onstar VCIM with an Onstar/Bluetooth VCIM but GM will not do it for you unless it's a warranty or defect replacement. The OS/BT Module is part #20783877 and the BT Antenna is #15938939.
If you have any luck finding and adding these modules let me know!