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".
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.