I got the same message this morning after a scan and was worried. I also have FIOS and the same router. But when a free antivirus program tells me to change system settings I draw the line. (I, and many others, had a HUGE issue with free Panda AV which effectively labeled all system .dll's as non legitimate and promptly REMOVED them!) So instead I called Verizon tech support and after 1-1/2 hours it was determined there was no threat to my router.
So all she did was reset the router to factory defaults on her side and I did the same thing on my side by pressing and holding the red reset button on the back of the router with a pen for 15 seconds. It took about 3 minutes for the router to come back up. I then went into the router settings by opening a browser window and entering 192.168.1.1, entering the router's default username and password located on the bottom of the router, and the settings page will open up.
She verified all the connected devices she saw on her end with what I was seeing on my end and said that no other unwanted connections were there. She assured me my router was working properly, my firewall was up and running and I had no security issues.
Also, I asked her if changing the DNS address to 8.8.8.8 like Avast's message said made sense and she said "no."
So bottom line, ignore the Avast warning. This one is on them.