I have a few questions about your installation. If you have Gigabit service, it would come from your ONT (where the optical fiber terminates) and then as an Ethernet cable to your router. It should be connected to the WAN port on the router. Is that the case? If none of the LAN ports have Ethernet cables in them, are you only using wireless connections for all the devices in the house? The router serves as the wireless access point for any wireless devices you have accessing the Internet. (Also if that's the case, Gigabit service is way overkill, but that's a different topic.)
If I am understanding what service you have, you need the Ethernet cable going into the WAN port regardless of where you locate the router. If you can run a new Ethernet cable from the ONT to where you want the router upstairs, that will work. If it's simpler, you can also use an Ethernet coupler to connect two cables together to run a new line upstairs. It's best to keep the overall run length from the ONT to the router around 100 ft or less.
Realize that you are moving the wireless access point upstairs, too, which will change the strength of the wireless connections. Those that were upstairs and perhaps a bit weak will probably get better, and those that are near where the router is now will likely get weaker. It may not make any difference in the performance you perceive, but it's worth mentioning.
If you don't have Fios TV service (and therefore, no Fios set-top boxes attached to TVs), I'm not sure that the coax cable is doing anything at all. Unless you have something like network extenders connected to coax elsewhere, you can likely leave that disconnected.