There is no direct connection from New London to Montauk. However, you can take the ferry to Montauk then take the taxi to Montauk. Alternatively, you can take the drive to New London, take the car ferry to Orient Point, then take the drive to Montauk.
From New London ferry to Montauk then taxi to Montauk. 1 h 6 min $79. From New London drive to New London, car ferry to Orient Point, then drive to Montauk. 3 h 11 min $49.