What do you mean by 'connect to the router'? Are you wanting to take the router to one of those sockets and plug it in there? If so, that may work.
If you're wanting to leave the router where it currently is and use the telephone extension like a LAN cable, then most likely the answer is no. You could potentially use the cabling if it's suitable but that's unlikely I would say and you'd need to change the socket from a BT phone socket to an RJ45 socket.