The situation is succinctly summed up by Ellicott:
The exact words “Out of his belly shall flow rivers of living water,”
are not found in any part of the Canonical Scriptures of the Old
Testament, and yet Christ Himself utters them with the formula of
quotation. This will be a difficulty only to those who value letter
and syllable above spirit and substance. It may be that the words
which our Lord actually uttered in the current language of Jerusalem
were nearer to the very words of some passage in the Old Testament
than they seem to be in the Greek form in which St. John has preserved
them to us. But it is instructive that the thought is that which our
Lord Himself, or St. John as representing Him, considers as the
essence of the quotation. The thought meets us again and again in the
Old Testament. See the following passages: Exodus 17:6; Numbers 20:11;
Psalm 114:8; Isaiah 44:3; Isaiah 55:1; Isaiah 58:11; Joel 2:23; Joel
3:18; Ezekiel 47:1; Ezekiel 47:12; Zechariah 13:1; Zechariah 14:8.
This frequent reference to the refreshment and life-giving power of
water is the more natural in the East, where drought is a fearful evil
ever to be guarded against, and a well of water a blessing always
sought for as the first necessity of life.
Thus, it appears that Jesus was quoting a common idea in the OT, and NOT a particular passage. The only places "living water" is mentioned is in places like Zech 14:8, Jer 2:13, 17:13, SS 4:15, etc. However, in none of these is there a mention of living water flowing out of a person. The closest we get is:
Jer 2:13 - “For My people have committed two evils: They have forsaken
Me, the fountain of living water, and they have dug their own
cisterns—broken cisterns that cannot hold water.
However, in this, the source of God not people. However, the NT teaching is that people were to imitate God and Jesus mentions this in another places as well:
John 4:13, 14 - Jesus said to her, “Everyone who drinks this water
will be thirsty again. But whoever drinks the water I give him will
never thirst. Indeed, the water I give him will become in him a fount
of water springing up to eternal life.”