No display after GPU swap

I just got back from campus—finally finished up with classes for a bit—so I brought my GPU home hoping to squeeze in some gaming, but I’m staring at a blank screen.
Basically, I have this ancient NVIDIA card that connects to my monitor via a VGA cable, which works fine. I tried swapping it out for a beefier Intel card that uses HDMI, but there's absolutely no signal. Interestingly, the old NVIDIA card actually has an HDMI port too, but even when I try using that, nothing happens. Does this mean I need to dive into the BIOS and toggle some setting for HDMI output? Just a side note: my monitor doesn't actually have an HDMI port, so I'm relying on one of those cheap converters, but I don't think that's the culprit here. Also, is there any chance the Intel card won't output anything because of leftover NVIDIA drivers hanging around in the system?
edit: To make matters worse, once I switch over to the HDMI connection, I can't even get into the BIOS because I can't see a single thing.
I tried switching the settings from onboard to PCI, but no luck. Should I try messing with the PCI Express settings instead?

Maybe you could list out some specifics so we don't have to go asking an oracle like AC Milan for info? Like, what motherboard are we talking about, which CPU, what was the old GPU, what did you try swapping in, and which version of macOS are you running? Also, is that 12V rail connected properly?
We aren't even sure if everything's actually working once you plug the old card back in—like, are you even getting a picture on the screen?
And just a heads-up, there isn't really an "HDMI" setting in the BIOS. Unless your motherboard has integrated graphics, the only thing you'll see is an option to toggle that onboard chip on, off, or set it to auto, along with how much RAM to carve out for it.

I was thinking about tinkering with the PCI Express settings in my BIOS—maybe try locking it down to see if it stabilizes things