Most MoCA adaptors are designed for MoCA LAN. The ONT uses a different channel for MoCA WAN. I'm not sure if MoCA WAN adaptors are available, but if you can find one, it should work.
That said, MoCA WAN is limited to 100Mbps. If you're on a higher speed tier, Verizon will be hesitant to provision the ONT for MoCA WAN. If you are on a lower speed tier and upgrade, they may want to send a tech to run the Ethernet cable and switch your ONT from the old BPON model to a GPON model.
Another option is to install a MoCA LAN adaptor at the ONT and another at your router. This only works if you do not have Verizon TV services. Of course, you'll need a power source for the adaptor, which could be a problem.
If you have FiOS TV services, you pretty much need one of their routers in the system. In that case, turn of it's WiFi and plug your router into the LAN port on the Verizon router. That's the easiest way to use your own router and maintain TV services.