Praise be to Allaah.
Holding such competitions is something good, if the aim is to encourage children to memorize the Book of Allaah from an early age. There is nothing wrong with encouraging them by offering prizes and gifts, and you will have a reward from Allaah. This has to do with cooperating in righteousness and piety, as Allaah says (interpretation of the meaning):
“… Help you one another in Al-Birr and Al-taqwa (virtue, righteousness and piety)…” [al-Maa;idah 5:2]
And Allaah knows best.