1. "I went to my network connections."

If this was on your machine, this is not where I would look. Hint: look on the router configuration pages.

2. It may sound odd, but I only give direction and if downloading a manual is required for me to step you through menus to find SMNP in the router then this is something I will NEVER do. Why? My outlook is it is your task to read the manual and learn the menus. Sure, I'll toss the occasional quick answer out but we are here to learn, not just get fish.

3. The SNMP item of concern is in the router. Research that.

4. "Do routers store data in them or do they just travel through?"

a. Some data is stored. Example? The assigned IP addresses and such. Recent DNS lookups are not unusual.
b. As to data travel through. Yes, and with SNMP on the router, a report is sent to wherever the router was configured to send it.

Dig in.

Bob