When I disconnect its power, I lose Fios internet. Is it possible to move it to a hidden place? Thanks
It is a power supply for a Verizon ONT. Look for where your Verizon fiber is connected to your house wiring via their box (ONT) or to a Verizon modem. You'll probably find a pair of power lines coming to it from your house phone wiring.
Usually the power supply would go next to or inside the ONT. Why it is installed here, we can guess: The Verizon fiber termination was wanted in a location where there is no power receptacle, so they ran power to it via your internal phone wiring. From there perhaps you use an ethernet cable to get to your router somewhere else. It seems odd but that's my guess.
Can you move it? As long as your ONT gets power somehow, you can move this or it.
Edit: a bit more advice: Stop using obsolete wiring for critical new services. In most homes these days the internal phone wiring is disused. If you have a home phone line at all, it usually serves only a portable phone base station. So when rooms get redecorated, any surface wiring or wall jacks like this one tend to be cut out and removed without thought to phone service. For that reason, you SHOULD, when convenient, move this power supply and the Verizon ONT to a location where they do not depend on your 1970s phone wires, but instead are connected in a more obvious and maintainable way.