AsRock offers the cheapest sub-ATX sized board that I'd actually try to run QHD at 60Hz on: the ASRock AM1H-ITX. Per AsRock spec, the DP port is 1.2 compliant, so it should be capable of QHD/60hz. Putting an AMD Athlon 5370 in the CPU slot should just about drive that resolution. Not sure I'd risk it with anything lower than that, but if you're the adventurous type, then you can get integrated, very cheap solutions like the ASRock QC5000-ITX/PH or the ASRock J3710-ITX (mind you, I'm not a huge AsRock fan, they just seem to be the ones offering the cheapest, most prolific models in this space. I'm guessing other brands offer solutions here, I just can't seem to find them).
FWIW, it seems that any FM2+, 1150, or 1151 motherboards with a DisplayPort out should all conform to at least DP 1.2, meaning they support your desired display. On the Intel side I'd tend to advise packages with HD 4400 or higher graphics solutions, but it is possible I'm way off base on that and plain old HD graphics will work just fine due to the magic of Intel's Quicksync decoder - but definitely do not try anything from AMD without VCE or from Intel without Quicksync. Those two technologies should be considered mandatory.