This thread might be more at home in the Graphics Cards -section, as you're indeed after a GPU. But here goes:
Have you tried without a descrete GPU? Because if you're not gaming, then the integrated graphics should be ok, as well. (Dismiss if your MB/CPU doesn't have iGPU.) only has vga output on the igpu and this sytem is dual monitor
Another angle, just in case:
When you tried with the GT610, you did use its own DVI port, right? YES Because when you plugged in the GT610, the BIOS probably disabled your iGPU, and at the same time its ports went dark, as well. Also, you did try changing the input via the monitor's OSD YES Or has it been on the same input and cable all the time, working normally earlier? Monitor has worked on same input but not in this pc
When you say it gives a "blank screen", does it show ANYTHING at the bootup? Because the Bios/Uefi bootup is probably 640x480/800x600/1024x768, or something universal like that. So it should probably show up, in any case. Is the rest of the computer booting up normally? Does the HDD seem like it's still loading the OS? Nothing on screen at all, yes it looks like the is booting up
Well, there's also the chance that the monitor lacks a scaler (which actually might be the case), and/or it might not want to show anything on certain resolutions with certain inputs. I think I once had a similar problem with VGA and a HDTV. The system stayed blank until it booted all the way to Windows/Linux. So I couldn't access BIOS, at all. Worked OK with HDMI and DVI-HDMI, though. And anyway, that would be quite an oversight with an actual COMPUTER MONITOR.
My opinion:
It's most probably NOT a case of limited resolution support. If the GPU/iGPU doesn't like 2560x1440, then it should just fall back to a lower resolution, or at least say something like "Out of range".
Also, is there some specific reason you want nVidia? Something like Radeon R5 230 or R7 240 should be cheaper. No reason just prefer Nvidia