Is it possible to use a GPU to process for an Onboard Graphics port?

My PC only has 2 HDMI ports, one in the video card (a Nvidia GTX 670) and one for the onboard graphics (Intel 4600). The HDMI port of my GTX 670 is feeding my TV for couch games I play with a controller (Saints Row, Sonic Generations, Gauntlet etc), while the onboard graphics is feeding my monitor. I would like to use my monitor for keyboard+mouse games like WoW, Civ V, and shooters.... But I figure I'm stuck with the limited graphics of my onboard chip for those, and yeah, hence trying to bridge the two adapters or something.

I guess I'm trying to have my cake and eat it too.

The GTX has 2 DV-I ports and a 4th port I've never really seen before (Skada? maybe it's a skada), though I only have one HDMI-DV-I adapter and it's on the monitor end. If the only solution is another adapter or a DV-I cable so be it, but I'm looking for something cheap until my next paycheck.

Specs: Alienware x51 R2 Windows 10 Intel i7-4770 CPU @ 3.40 GHz 3.40 GHz GeForce GTX 670 16GB RAM 1.80 TB HDD