If I’ve bought Apple TV in Europe will it work in North America?

yes, of cause you will need to get a new cable (it's an industrial std. my old playstation 2 and cd player use the same cable any AV store should carry them and likely some supermarkets)

or use a eu to us plug converter

Apple TV's work anywhere there is Apple TV service. You may need a new power cable or a simple adapter.

Your welcome.