The HDMI on that monitor won't do 1920x1080 144 Hz. DisplayPort can, and DVI-D Dual Link can. However, in order to get GSync to work, you will need to use DisplayPort (1.2+).
HDMI is more of an A/V cable, as DisplayPort is useless in that environment. It's missing features HDMI got, 3M max range for DisplayPort wouldn't work well as an A/V cable standard, as opposed to HDMI's 15M. And, the frame packing which is mandatory for 3D to work on TV's, is HDMI exclusive, DisplayPort can't do it.
However, because DisplayPort and HDMI are both using active signals, an adapter will work, but you won't get all the features. If it's purely for compatibility, then use an adapter. If not, then always go with DisplayPort when possible, DVI-D Dual Link if you have to, as monitor is still using HDMI 1.4.
If you use an DisplayPort to HDMI adapter, or vice versa, what's going to happen is that it's going to be limited by the weakest link, in this case HDMI, regardless of which part of the chain it's in.
All the best!