This problem has been driving me crazy. We've had FiOS for the past 17 months and I've had this issue that whole time. Usually once a day, but sometimes several times per day I will lose DNS. I can ping numeric IPs fine (like 126.96.36.199) but if I run tracert or ping a DNS address such as www.verizon.com it will fail saying there is no DNS. Also I can't access any websites in my browser when this happens. Sometimes restarting the router works, but usually I have to restart the router AND reboot my PC to get DNS working again. I have the same slim black router I got when we first signed up. I've changed no settings in the router, nor in Windows 10. I have an ethernet cable going from my PC's network card to the router. I should mention that I've gone as long as 1 month where DNS worked fine, but it always fails again and this past week it's failed at least 3-4 different days. I hope there is a solution to my problem
If it's all devices, login to the router and make sure it's DNS settings are correct. The default is to pull DNS settings from the WAN DHCP query. You can change DNS on the router to whatever you want. Google's DNS and OpenDNS are popular, fast services.
You can also change DNS settings on your computer and have it ignore DNS as sent by the router. This can be a good way to diagnose. Of course, any DNS changes on your PC don't do anything for other devices on the network. Again, I'd use Google's DNS or OpenDNS for this.
If it's just on the one PC, it's likely it was set to use a particular DNS service. Find out what that is and investigate.
One other note, traceroute on Windows with FiOS doesn't work well. You'll only see one hop with the built in Windows traceroute. This is because Windows traceroute uses ICMP packets and the FiOS network "eats" them. Linux and Mac traceroute use UDP packets. You can find 3rd party tools for Windows that use UDP packets. PingPlotter Pro is one such tool.
Since it's only the PC, it's likely it's network settings are messed up. If you google "Windows 10 DNS settings" you find many resources that step you through the process. If you are using a different OS, you can find tutorials for those, too.