Are there any religious texts that prohibit leaving a newborn child alone in a room, or that command placing a copy of the Quran next to them?

There is nothing in the Sharia that indicates placing a Mushaf (copy of the Quran) near a newborn if left alone. This might be a custom of some peoples based on fear of jinn and devils. Protection from this comes through reliance on Allah, seeking refuge in Him, and reciting the narrated supplications. It should be noted that leaving a child alone is not advisable due to the potential harm it exposes them to. There is no harm in placing a Mushaf in the newborn's or the mother's room, provided it is in a respectful, clean, and safe place.