Well, after scouring the internet in an attempt to figure a solution for myself I have finally admitted defeat and here I am.
The issue is hard to pinpoint as it happens at random. Be it gaming, browsing the net, or even opening a window to navigate around my computer. It is always the same. The stop error reads as 0x000000 with no other indicators. Thankfully, through checking with a program called BlueScreenView, I was able to isolate the culprits and get a general idea of what is causing this.
Keep in mind I have already tested the RAM thoroughly, switched hard drives, fully reformatted, and even installing a fresh installation of Windows 7...multiple times. This has happened on two separate motherboards with varied processors as well. This leaves me assuming that it is the graphics card.
I will include three minidumps. The first only shows the ntoskrnl.exe as the issue, the second ties it in with nvlddmkm.sys, and the third ties it in with pacer.sys.