Your first problem is the lan port on your router is actually running as a switch with a single port connect to the router chip. In many cases the lan ports on a switch all appear as a single port to the router.
Now "maybe" the switch chip in the router can show you what is called the mac forwarding table. This is a memory that keeps track of what mac addresses the switch has seen on each port. Note this is mac address not IP address.
So if you hook a PC to a lan port on the router directly you should only see 1 mac address on that port. If you were to see multiple mac addresses you could assume that there was a switch connected to that port that allowed multiple devices beyond it.
This is only mostly true since there are devices that can have multiple mac addresses.
So it would let you know what port on your router has a switch but it does not let you know what is connected to what port on that switch. You would need a switch that has some method to get access and then display the table. Most simple unmanged switches even though they have a mac address forwarding table give you no way to see it.
Some commercial equipment has the ability for example to limit what mac addresses are on certain ports or only allow certain mac addresses after you authenticate/login to a server.