Well, that pretty well rules out anything specific to Windows, then. It's not your drivers or anything like that.
Unfortunately, that doesn't leave a whole lot on the table other than hardware failure. If the system boots with the card in place, then it seems like there can't be anything
catastrophically wrong with the card. That is, if there were a serious GPU failure, you'd generally expect the system to crash, even if though the device is "peripheral". Perhaps only the card's RAMDAC has failed? It would be helpful if we could gather more information about the failure, but that could be tricky if you can't get some kind of two-way communication with Windows.
If it
is a hardware failure, it could occur about anywhere along the line. You've already tested the monitor separately on another computer; that's good. We also need to eliminate the video cable, if that wasn't already tested with the monitor. There could be something wrong with the motherboard's expansion slot for the video card -- PCI-E (express), I'm guessing? Or the problem could be with the card itself. Either way, it would be helpful to know if the card is even being detected by BIOS and Windows.
If you've set up your desktop PC to allow for remote desktop, you could talk to Windows that way from your laptop and see what the device manager is showing. If you didn't enable it before this happened, it gets kind of complicated. However,
it is possible to remotely enable remote desktop.
Another thing to try (if you have a PCI-E bus) would be to put your video card in a different PCI-E slot. The card should be able to negotiate down to a smaller number of lanes if you plug it into an x8 or x4 slot. I'm not sure if they'll fit at all into x1 and x2 slots. I suspect not. But if the card will plug into a slot at all (even with left over pins on the card), it is safe to do so. Doing so will kill your performance, of course, but it could be diagnostically useful if you can get the card to work in a different slot. If that's not an option, I'd try plugging it in to a completely different computer, if possible.