Unfortunately, you need the Displayport on the motherboard's I/O in order for this to work. To my knowledge, even if you used an HDMI converter, it would only to be to a single link DVI, which wouldn't be enough to support your 1440p monitor. You could still use low end Nvidia cards, like this one:
However, as you can see, they are not so cheap. The other route (if you want to keep using Gigabyte boards) may be to change to a 1440p monitor that has an HDMI output such as this one:
I was in the same situation as you just a few weeks ago, and I could not find a satisfactory Gigabyte board that had DP either - at least not in Mini ITX form factor. Hence why I ended up buying a Mac Mini instead, and it's actually what I am using right at this moment. The only downside for me was that my Achieva Shimian 1440p monitor has the one Dual-Link DVI output, so I had to buy that expensive Apple Mini Displayport to Dual-Link DVI active adapter in order for it to work.
Seems like no matter what, there is always a compromise (and an extra expense) when going to the other side.