I would make the nest to the near tree in a safe place that way nothing will happen to it.

if you are in the UK its an offence to interfere with a birds nest.

leave it to nature.

So a pigeon made a nest (if you can call a bunch of twigs on the floor a nest) and laid it's egg in it. We go out there all the time, so I guess it's not such a smart idea for it to be there.

How and where should I move it? Is the nearest window seal a good option?

I've put up a video of it's exact location on the balcony: https://www.youtube.com/watch?v=Fl7R25mIs20