Your problem is where does it log it to. Most network equipment has no storage device so the best it can do is hold some information in ram.
You are going to need a pc like device to store and likely analyze the data. The easier way is to replace the router and run a firewall/router software on a pc like pfsense. Some router have the ability to send data to a remote server. The one cisco commercial routers use is called netflow but there are third party firmwares for consumer routers that have a very similar function.
The largest issue with doing this is all the data is likely now encrypted. You can not see the actual URL anymore. You at best see the ip address but many sites use hosting services so the IP do not always...