I've read that for a lot of 120Hz, 144Hz and higher refresh rate, you have to use DisplayPort output from the GPU, and sometimes even 2 at the same time if the resolution is very high. But if I'm looking for a FullHD monitor with 144Hz refresh rate, can I use HDMI? Or does it depend on each monitor's model?
It depends on the HDMI version. You'll need HDMI 1.3 or above for 1080p and 120 Hz or more.
EDIT: Your monitor and graphics card do still need to support a 120 or 144 Hz refresh rate at the resolution you desire. DisplayPort works too.
The Wikipedia article has a really useful table: