No. The ethernet, E-SATA, and USB ports are not active on any of the current Verizon STBs (although the USB has power that could be used to re-charge a device).
You would either need a direct ethernet cable from your router, a wireless adapter / access point, or a MoCA bridge device (NIM-100 from E-BAY, an Actiontec product avaialable on Amazon, or an additional Actiontec or Westell router that is configured to be bridged). The NIM-100 has been discontinued by Motorola and there have been reports of dead ones in some recent purchases from E-Bay. There are instructions at dslreports.com on various bridging / router setups for FiOS..
BTW - this question has been answered many times and a search should have provided you with an answer. Remember, search first ask later 