Passing connection for multiple classes/methods
It may make sense to organize your objects in a different way.  With Read and Write extending Connect, you are basically treating them as if they are connections rather than being operations performed over a connection.

I think the most straightforward way to model operations over a connection would be something like this:

public static class GameConnection {
    private TCPMasterConnection conn;

    public void connectTo(String ipAddress, int port) throws IOException {
        conn = new TCPMasterConnection(InetAddress.getByName(ipAddress));
        conn.setPort(port);
        conn.connect();
    }

    public int[] read(int unitID, int startAddress, int count) throws IOException {
        // Do the read over this.conn
    }

    public int[] write(int unitID, int startAddress, int dataToWrite, int count) throws IOException {
        // Do the write over this.conn
    }

    public void close() throws IOException {
        conn.close();
    }
}


To use it, you then just

    GameConnection conn = new GameConnection();

    conn.connectTo("127.0.0.1", 8080);
    int[] readStuff = conn.read(1, 2, 3);
    int[] writeStuff = conn.write(1, 2, 3, 4);
    doSomeOtherThingsWithIt(conn);
    conn.close();


For more abstraction, you could make the GameConnection an interface and have different implementations for different transport protocols.