In simple terms, no.
The socket you have on the car is the 12N (the black one) which supplies the running lights on a trailer or caravan.
To use the second plug, the 12S (the Grey one) you will need to supply a 12S socket for the car and wire it in, it supplies the power to the fridge and battery charger when the car engine is running, and also the reversing light on the caravan.
You do not actually need to convert to 13 pin, but is much more preferable to the two 7 pin sockets which ,particularly if running the fridge when towing, where there are high instances of centre pin burn out with the 7 pin socket.