All bridges into NYC from New Jersey are toll bridges, and the MTA collects tolls on the major bridges and tunnels between the boroughs. But if you look at a map of New York City you’ll find plenty of bridges serving local streets where no tolls are collected. The Brooklyn Bridge, for example, has no toll. Neither does the Manhattan Bridge or the Williamsburg Bridge. Most of the bridges between Manhattan and the Bronx have no tolls, either.
So basically any bridge connecting to another state is toll. I know the bronx is part of NYS.