This issue is mostly caused by the power source.
Your PC might still run, the fan from the power source might still run, but the monitors will stay black if there is not enough power generated by the power source. You won’t be able to access even the BIOS.
This can happen to old power sources, 3-4 years old.
Another thing that you can check is to switch the monitor cable to the integrated video card, if you have such thing. But usually if your PC has worked before from the dedicated video card jack and now is not, most definitely is the power source.
Other things. If you hear beeps, check the RAM memory. If nothing works, you can eventually try to reset the CMOS (the BIOS jumper). Check your motherboard’s manual in order to locate the CMOS jumper.
But overall, the first thing that you need to do is to try to replace the power source and see if the monitors will run with a different power source.