The Quran is pleasant to hear and even a person who does not understand its meaning, like a child or a non-Arab, may enjoy listening to it.
Therefore, it is permissible to play the Quran for a child, especially if he calms down when listening to it, because it is lawful to recite Athkaar (the legislated supplications and mention of Allaah) for children.
The effect of the Quran to children and infants is something that has been observed. This should not be of any surprise because the Quran is a light, guidance and healing.
Allah Knows best.