Is there a quote from Jesus where he speaks about God's law written on the heart?
I don't think Jesus says it directly with the same words. But the same idea comes from how person should be born anew, become righteous and get the Holy Spirit. When that happens, person has the law written in his "heart". Here are the scriptures that tells how and what it means:
I will pray to the Father, and he will give you another Counselor, [Greek Parakleton: Counselor, Helper, Intercessor, Advocate, and Comfortor.] that he may be with you forever,- the Spirit of truth, whom the world can't receive; for it doesn't see him, neither knows him. You know him, for he lives with you, and will be in you.
John 14:16-17
But the Counselor, the Holy Spirit, whom the Father will send in my name, he will teach you all things, and will remind you of all that I said to you.
John 14:26
When the Counselor [Greek Parakletos: Counselor, Helper, Advocate, Intercessor, and Comfortor.] has come, whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father, he will testify about me.
John 15:26
However when he, the Spirit of truth, has come, he will guide you into all truth, for he will not speak from himself; but whatever he hears, he will speak. He will declare to you things that are coming.
John 16:13
Jesus answered him, "Most assuredly, I tell you, unless one is born anew, he can't see the Kingdom of God." Nicodemus said to him, "How can a man be born when he is old? Can he enter a second time into his mother's womb, and be born?" Jesus answered, "Most assuredly I tell you, unless one is born of water and spirit, he can't enter into the Kingdom of God! That which is born of the flesh is flesh. That which is born of the Spirit is spirit. Don't marvel that I said to you, 'You must be born anew.'
John 3:3-7
But as many as received him, to them he gave the right to become God's children, to those who believe in his name: who were born not of blood, nor of the will of the flesh, nor of the will of man, but of God.
John 1:12-13
It is the spirit who gives life. The flesh profits nothing. The words that I speak to you are spirit, and are life.
John 6:63
He who does righteousness is righteous, even as he is righteous. He who sins is of the devil, for the devil has been sinning from the beginning. To this end the Son of God was revealed, that he might destroy the works of the devil. Whoever is born of God doesn't commit sin, because his seed remains in him; and he can't sin, because he is born of God. In this the children of God are revealed, and the children of the devil. Whoever doesn't do righteousness is not of God, neither is he who doesn't love his brother.
1 John 3:7-10