The following verses contain every time the KJV uses the phrase “believe on.”
John 1:12 But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:
John 6:29 Jesus answered and said unto them, This is the work of God, that ye believe on him whom he hath sent.
John 7:39 (But this spake he of the Spirit, which they that believe on him should receive: for the Holy Ghost was not yet given; because that Jesus was not yet glorified.)
John 9:35-36 Jesus heard that they had cast him […]