I'd go with either
Free Time
time available for hobbies and other activities that you enjoy Free Dictonary
another definition
Free time is time spent away from business, work, job hunting, domestic chores and education. Wikipedia
Or if wanting more formal then:
Lesuire time
time available for ease and relaxation; "his job left him little leisure" Free Dictonary 2
I'll go with downtime
time spent not working; free or leisure time
Webster's New World College Dictionary
I'm looking for a word to describe the person you prefer to spend most of your time with who could be anyone from a relative to a lover. They’re not just a “companion”, they’re your “chosen companion” among many, if that makes sense. You have other people in your life that you care about, but you always choose to be with this person over everyone else, you always find yourself gravitating or coming back to them, specifically when "spending leisure time together with someone, doing any kind of activity", they're always your No.1 choice.
What can we call that person?
Note: this was also posted on r/whatstheword
The sentence is slightly unnatural.
"spending quality time" generally needs to specify who the other people are. You can sometimes use "at" too, as in "Spend quality time at home". The phrase refers to an enjoyable experience shared with other people or in some place, so leaving out those details makes it sound a bit strange. If your intended meaning is to say that spending time together will improve family relationships, you could rephrase it as follows:
My vision is to bring the family together by having them spend quality time with each other.
If you intend to say that the quality time will be a consequence of having the family members being in more proximity to each other than usual, it would be:
My vision is to bring the family together, thereby allowing them to spend quality time with each other.
It basically depends on what the exact intent of your original sentence is. But it will usually need a "with" and whoever or whatever goes with that "with".
EDIT
Perhaps the phrase "bring the family together" is also introducing ambiguity. It can mean getting the people to share the same physical space, or it could mean the figurative sense of improving the interpersonal relationships. I would also consider clarifying that phrase.
It's not entirely clear what you are trying to say. It's as though something is missing from your sentence.
Do you mean that spending quality time is the means by which you wish to bring the family together?
Or do you mean that you want to enable the family to spend quality time together?
It would be clearer if you wrote:
My vision is to assist the family to spend quality time together.