Sometimes the G fare shows on the website, but most often it does not. However, putting the reservation on hold and calling Meeting Services is, I've found, a good idea. If it's on hold, you can give them the record locator, and they can, if it is a G fare on hold, ticket it immediately with the upgrade certificate. If it is lower than a G fare, they can upfare the reservation and ticket it for you with the upgrade certificate applied. In both cases, IME, they actually do something in the computer to hold/claim the C inventory.
If you send the reservation to ticketing before calling Meeting Services, they may not be able to do anything with it until the normal ticketing process runs its course. In other words, if it's in Ticket Pending status, they often won't be able to touch it until the ticket is issued.