There are several examples that would indicate that Jesus could read and write. When he was in the wilderness tempted by the devil, he replied to his temptations, “it is written.” How did he know if he had not read it. Also when he addressed the Sadducees, about marriage in heaven, he said “ye do err not knowing the scriptures, nor the power of God.” He would have not said that if he himself was unaware of what the scriptures said. When the man approached him who asked Jesus how he could inherit eternal life, he asked the man, “what is written in the law, how do you read it? (Lu. 10:26)” Then Jesus told him, “You have answered correctly.” How did he know this if he had not read it?
Concerning his ability to write, he does show us stylistic skills and knowledge in writing. Concerning prayer, he said, “ask and it shall be given, seek and ye shall find, knock and the door shall be opened to you” (Matt. 7:7). He uses acrostic style of writing in this verse for Ask begins with an A, Seek begins with an S, and Knock begins with a K...which spells ASK. The greek meaning in this verse is exactly that, Ask. The whole theme of this text is to ask, and Jesus uses acrostics to both identify and explain it’s meaning.
He also used parables to teach, and this also was a another form of writing and speaking in order to reveal a hidden meaning in the story. In Luke 20:17 Jesus replies after telling a certain parable “And he beheld them, and said, What is this then that is written, The stone which the builders rejected, the same is become the head of the corner?” (Lu. 20:17 NIV). How did he know that this was written in scripture if he had not read it.
I am sure that there are more examples, but these just come to mind off the top of my head, I hope this helps.