We are hoping this is not a problem with a defective graphics chip because on a laptop that is a big problem.
The way you install the graphics drivers is important.
The AMD graphics driver package on that HP website
contains both the Intel and AMD drivers in one package. You only need to install that one package. These drivers are configured specifically for your laptop to use the switchable graphics feature. You do not need to install the 2 drivers separately, and that may be the cause of your issue.
Before you install new drivers it is important to uninstall the current drivers. Post back if you need more on this procedure.
You should also check this out: HP Pavilion g6-1000 Notebook PC series*-* Video, monitor - HP Business Support Center
And see if your issue is listed and what HP has to say about a solution.