I am seeing the messages over and over (thousands of times) in my FIOS router's security log (see photo). I feel like something is performing a brute force attack. How can I prevent this?
Unless you put a firewall in front of router, you can't.
Any device that connects directly to internet is subject to these attacks.
Make sure that Remote Administration via HTTP and HTTPs, as well as SSH and Telnet access are disabled in your router. The settings for these are in the Advanced > Remote Administration section.
Also, find and disable the Universal Plug & Play feature of the router. Then go to Firewall > Port Forwarding, and clear out any port forwards which you don't know about. Ones which you can't delete are likely going to be set by Verizon for various services like Remote DVR to work correctly.