Can I connect a router via ethernet cable off of the new Telus modem?

If it's the first picture, plugging one of the Deco nodes or another router into one of the LAN ports on there should work.

There are two things that right off that bat could be an issue, and the second I know for sure affects the Deco.

1. If you're connecting the LAN ports on the Telus gateway to the one of the LAN ports on the Deco / third party router, that would cause an IP conflict and a failed connection. Normally if it's connected from LAN on the gateway to the Internet or WAN port on the back of either of those devices, it should work.
2. IP address subnet conflict. I know the Deco will be affected by this as the default address is 192.168.1.1. You'd need to make sure the Deco / router is not using the same 192.168.1.x IP range that the Telus gateway is by default. You'd have to log in to either the Deco or third party router and change the IP range from 192.168.1.x to something like 192.168.10.x. You may have to make these changes to them while they are not connected to the Telus gateway. Once changed, reconnect the Deco / router to the gateway and reboot them. Check if things work.