What Edg1 is proposing
Unfortunately that is incorrect. The FiOS Network Adapter works on MoCA LAN which operates at a different frequency than MoCA WAN from the ONT.
There are two types of MoCA. MoCA WAN from the ONT and MoCA LAN which the adapter is using. You can do what you are try to do but you will need another Network adapter and have Verizon switch your connection to ethernet at the ONT.
You need to wire like this...
ONT> ethernet > 1st FNA > coax > 2nd FNA > ethernet to your router.
should work so long as you are not subscribed to Fios TV service. If you always planning on using Ethernet and your own router, I'm going to guess you are not a Fios TV subscriber. If you are and your coax is all connected within your house (which is pretty typical), you would have an issue with getting video on demand and guide info, which is (also) sent over the MoCA LAN frequencies to Verizon's set-top boxes. The Fios Network Adapter does a job similar to the pair of ActionTec ECB6200s that I bought to do a similar job in my house, and it costs less ($110 for a pair rather than $170 for a pair). The Verizon FNAs claim bonded MoCA, so they should be capable of ~1 Gb/s speed, too.
My setup: 75/75 Mbps; Linux router (2x 1Gbps ports) + iptables; Linksys WRT1900ACS in Bridge Mode]