In my T610, i installed a XFX GeForce graphics card which i modded to fit in a x8 slot and then within BIOS, set the "Embedded Video Controller" to "disabled". Upon connecting the monitor to the XFX card, i could the view the machine booting up and windows 7 had no issues with aero interface (the onboard matrox g200 would not work well with aero).
So I would be very surprised if the T620 could not handle a PCIe video card and in fact I understood that one could order Quadro cards with their systems which were a supported option. What you may have is some conflict occurring when you use a graphics card in combination with a GPGPU card. If the T620 is not a production machine, i would remove the GPGPU card and drivers and install a graphics card, set the BIOS and connect the monitor to the graphics card. If it works, i would install the GPGPU card again and see if they both work.
Maybe the bods at NVidia can shed some light on this issue as too often you on this forum one hears 'not supported'.