This question comes up far more frequently then I would have expected – and the answer is, unfortunately, generally no.
In the 1600s, there were many more colonies in New England than states existing today. Much of Fairfield County fell under the political jurisdiction of New Haven Colony and the commercial realm of New Netherlands. Neither colony had similar political or social views to Plymouth. In typical settings, they generally would not have interacted.
Remember, ancestors don’t have to arrive on the Mayflower to have a fascinating history!