I think you do not want to use the speakers from the monitor so HDMI or DVI eider one will do.
The quality of the image is about the same.
If you want to use the HDMI port you need a HDMI cable, if not, just use the DVI cable that came with the monitor.
I have 2 monitors, one connected with DVI and the other one witch is LCD TV with HDMI cable.
The HDMI cable is not that expensive if you want to buy one .
"I was wondering if I need to buy a separate HDMI cable to connect my monitor to my graphics card or if I use the DVI cable the monitor comes with will I be able to get HD quality from that? " -short answer, yes!
EDIT: HD refers to video having resolution substantially higher than traditional displays. The video card can support a much higher resolution than 1080p(1920x1080) with a digital signal(meaning HDMI or DVI).
HD refers to anything above 640x480. In the TV world, this made for a huge opportunity to sell new LCD/Plasma/etc TV's that were capable of higher resolutions which they designated at 720i/p & 1080i/p. In the PC world, LCD's have been doing "HD" for years. Even CRTs were doing HD, people just weren't noticing as much because large form factor CRTs didn't get common and popular enough before LCD's took over.
Anyway -- short answer is Yes as ionut19 said, DVI or HDMI, either is good, you'll get the same quality. The major difference is simply that HDMI bundles full quality audio along with full quality video, which DVI does only video and you'd need a separate connection for audio, if you were outputting to a TV without a sound-system. Or, if outputting through a sound-system, it would need to be able to pass video through to the TV. But since this is a computer monitor in question, the audio portion is irrelevant.