Did the questioner break his oath by entering the Quran memorization class before the time he specified in his oath?

You will not break this oath, because your oath was to enter the memorization program next year, unless your intention was not to enter before then. If you intended to make your entry contingent upon God's will, then you will not break the oath. However, if the mention of God's will was merely for blessing, then you will break the oath and expiation is due upon you.