You seem to be confusing the WAN and LAN connections.
With FiOS, the WAN connection is either MoCA or Ethernet, but not both. This is how the router connects to the ONT. To change it, you must call Verizon. They will configure the ONT to provide the internet connection of the requested media.
On the other hand, the LAN runs both MoCA and Ethernet at the same time. To use Ethernet, simply plug an Ethernet cable into the LAN Ethernet jacks on the router. You can add a switch if the four ports on the router are not enough. To use MoCA LAN, simply connect your coax network to the coax terminal.
In short, you probably don't have to call Verizon at all. Just plug your new Ethernet wiring into the router.
If you still have questions, tell us more details about the equipment you currently have, how it is connected, and what you want to do.
Enjoy,