"Quality time" is time that you spend with your family members, someone you're in a relationship with, or really close friends. It's time that you spend doing things together, like talking, playing sports together, or going out somewhere together. If you're talking to a family member and watching TV or doing work at the same time, that's not "quality time". You have to be able to concentrate on the other person.
In American culture, family members are expected to "spend quality time" with each other. If parents don't spend quality time with their children, the children may feel like the parents don't love them. Husbands and wives, as well as serious boyfriends and girlfriends, are also supposed to spend quality time with each other.
It's a common New Year's resolution to say that you want to spend more time with your friends and family.