All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
If you are asking for a special supplication to memorize the Quran, then the answer is that there is no specific supplication in this regard.
Fatwa Noorun ‘ala-Ad-Darb of Shaykh Ibn 'Uthaymeen reads:
“'What is the best supplication to memorize the Quran, and what is the way which you advice for the memorization of the Book of Allah; may Allah reward you?'
'I do not know a supplication for the memorization of the Quran, but the way to memorize it is that the person accustoms himself to memorizing it in a regular manner. There are two methods in memorizing it:
The first method:
To memorize verse by verse, or two verses by two verses, or three verses by three verses, according to the length of the verses.
The second method:
To memorize page by page. People differ, some memorize it page by page, meaning he repeats reciting the page until he memorizes it. Some of them prefer to repeat the recitation of a verse until he memorizes it, and then he moves to the second verse, and so forth. So he continues to do this until he completes the memorization of the page. Also, regardless of the method that he adopts in the memorization, whether it is the first method or the second method, he should not exceed the memorization of a verse until he masters it so that he will have a strong foundation, and he should review what he had learned every day, especially in the morning, and when he knows that he mastered what he had memorized, then he may move to the second lesson (page).'”
Among the supplications which you may say in this regard is: “O Allah, make the Quran the Spring of our hearts, the light of our chests, the remover of our sadness and the reliever of our distress. O Allah, teach us what we do not know of it, and remind us of what we forget from it, and enable us to recite it during the day and during the night according to the manner that makes You pleased with us.”
You may also say, “O Allah facilitate for me the memorization of your book, and grant me its knowledge, and teach me its interpretation, and bless me by making me act upon it.”
Allah knows best.