Yes, you can use your Verizon router with spectrum internet. To use the service, you don’t even need any new equipment or wires, the only thing you need to do is just attach your router and turn on your service.
As you already know Verizon FIOS does not use any modems, you will be getting your internet service directly from Spectrum.
To complete the process your Verizon FIOS router will add in an Optical Network Terminal (ONT). From the ONT, you will see a TV wire to connect with the Router you have.
As you will be using this cable only for the internet then you can switch your router in Bridge Mode. When you are in Bridge Mode, it will not perform processes like Network Access Translation (NAT). Well, turning on the Bridge Mode feature varies from router to router you will be using.
See how to set up the Bridge Mode on a Verizon Fios G1100 Router
If your Verizon router is not compatible or if you are having any kind of problems with the internet then contact customer support and get a new router or rent one.