does strong WiFi improve phone signal quality?

No, what does WiFi have to do with making a phone call through your mobile carrier (Verizon, Sprint, AT&T, etc). What determines your phone signal strength is your relationship to the nearest cell tower.

You should know however, in areas where tower interference occurs, for example, remote area's, or areas near the water (homes on cliff's or bluffs near water), some have obtained a cell booster that plugs into one's internet router to boost the cell signal. Essentially, this allows these residents to connect to their mobile carrier through the internet and mimic a cell tower. I have a friend who lives on the water and has AT&T, he cannot get any service in his house. AT&T set up a box that plugged into his Internet router and allowed him to get At&T service in his home.

Unfortunately, Mobile Carriers have recommended these boosters so they don't lose your business. Truth is, towers can't reach everywhere efficiently. If you have friends that have a different carrier than you, I would have them come over and see if they have more service than you. If so, I'd consider changing your carrier.