Is it possible to to get my network to work off of the Ethernet port on the Verizon hardware installed at the fiber box rather than my Coax. I find that if I don't run the coax to the point of the router first that I have issues with Fios TV picture. I would think this is a better set up and enable faster internet.
You can but ...
1. You will still require the Coax to be run to the router because it provides both the WAN (when on coax) and LAN (which the STB's use to communicate to the internet for guide data, etc.). The router bridges the MoCA LAN to the local LAN network. Coax being broadband carries the internet at around 1100Mhz and the local LAN around 1000Mhz (or possibly the other way around).
2. The bandwidth on the Coax over MoCA is upwards of 200mb/s (twice that of which is available on the standard 100mb/s interface of the WAN ethernet connection) -- both the Coax and Ethernet connection methods are substantially faster than any of the packages present in the home installation unless you have a 150mb package -- which is always done over ethernet anyhow because you need the GigE speeds that particular router/ONT combo can give.
If you're have picture quality issues based on the order in which you run connections to your TV's and router, then check for bad cables and splitters -- particularly older splitters that aren't rated for digital tranmissions.
Changing the WAN connection method has to be done by tech support.