I'm not an expert on Androids so I can only give basic advice; from experience, it seems to me that it really depends on the phone itself, as every manufacturer will distribute their own "versions" of each Android build. While you could try rooting the phone, there's no guarantee you can make Android 4.4 itself compatible as it seems like most roots are built off of the stock versions distributed by each manufacturer.
Unfortunately, it looks like the ZTE Quartz is currently limited to Android 4.3. You may wish to find some kind of Updater/ZTE app hidden on your phone somewhere, or contact their support line to check if it's possible to upgrade or if a new version is on the way.
EDIT: I think in the Phone Settings menu, there's usually a "Software Update" option. That might be a good place to check.
