Hello @johntimber, and welcome to the Forum!
There are a few neighborhoods which might fit the “car-free” requirement: 1) Sandy Bay, West End, and there are a few neighborhoods outside of Coxen which would give you quiet you are after.
My first thought, however, was to move east if you’re looking for true silence. Oakridge and Jonesville are two good areas with easy transportation access to French Harbour.
When are you are on the island next?