H61, should be fine with a modern GPU.
An OEM board however, is a tough one to judge - you may well find a lack of BIOS support.
You will likely have to disable "secure boot" in the BIOS, to have any chance.
Unfortunately, the only way to know is to try.
If you have access to a new(er) GPU that you could try temporarily, you should be able to find out pretty easily.
If something like an R9 era card from AMD, or a 700 series from Nvidia work, then the 580 should also.