If you're hoping to get an extra FIOS router to plug into coax cable in your room, that's not going to work. There can be only a single cable mode/FIOS router per household, unless you're paying for a business account and requested multiple IP addresses. Residential accounts are limited to a single router.
If you're planning to run an ethernet cable from the existing FIOS router to your room, and plug a new router into that for closer WiFi, yes you can do that. But you have to first convert the new router into an access point (unless you want this to be a personal WiFi network isolated from everything else in the house). There's a sticky which explains how to do this. (Although the easier route would be to simply plug that ethernet cable into your computer).
http://www.tomshardware.com/forum/36406-43-convert-wireless-router-wireless-access-point
The little "whip" omnidirectional antennas and the antennas built into router boxes are only omnidirectional in the horizontal plane. Their signal strength forms a torus (donut shape), and have very little signal strength up/down. If your home is more vertical than horizontal (like some townhouses), then you could try putting the FIOS router on its side so the majority of its signal strength is going up/down. Rotate it so the top/bottom (when upright) correspond to the horizontal directions where you need the least signal. This may allow you to get a better signal without having to run wires or buy new equipment.
http://mpantenna.wpengine.netdna-cdn.com/wp-content/uploads/2015/02/FIGURE-1.png
If that doesn't work, I would suggest getting a router with at least 3 external antennas. Either set it up as an access point as above. Or plug its WAN port into the FIOS router with a static IP address, disable WiFi on the FIOS router, and set its DMZ to the new router's WAN IP address. At that point, you can pretend the FIOS router doesn't exist and just use the new router to control everything on your network. Orient one of the antennas horizontally so its signal strength torus intercepts upstairs and downstairs. Or orient two of the antennas at 45 degrees (opposite each other) so their toruses are not perfectly horizontal but give better coverage upstairs and downstairs. If your devices have external antennas, they should be oriented to be as close as possible to parallel to one of the antennas on the router.