While there is no specific record of Jonah asking God for forgiveness, but his actions seem to imply a repentant heart.
The message of the book of Jonah is that God has compassion on all of mankind as well as the whole creation. It’s not really all about Jonah, but we can get some lessons from his experiences.
Jonah was a prophet who tried to run away when he didn’t like the assignment he was given. He was to tell the people of Nineveh to repent, but fled in the opposite direction. When the ship he was sailing and overtaken in a violent storm, (Jonah 1:9-12), he admitted to the sailors that he was responsible for the calamity. Jonah’s confession might be viewed as an act of contrition or repentance on Jonah’s part. To save the others, he acknowledged his error and was cast into the sea.
Once inside the fish, Jonah reflected on his behavior and also on the Lord’s mercy. He realized that he had “sunk to the depths”, but was confident that the Lord still heard his prayers (2:7). In 2:9 he vows to “make good”, acknowledging the Lord’s salvation. This could also be considered an admission of guilt or a form of repentance. It was only then that the fish was commanded to deliver him to dry land.
The Lord asked him to go to Nineveh again, and he obeyed. His message was heeded by the King and the people. When Jonah changed his conduct and followed the Lord’s direction, his message caused the people of the city to turn from their evil and violence. God saw their repentance and did not bring the promised destruction upon them.
In 4:1, Jonah is angry that the Lord had compassion on the people of Nineveh, and the Lord questions his “right” to be angry. While Jonah waited to see what would happen to the city, God provided relief and comfort with the miracle vine, but then withdrew it. When Jonah felt “angry enough to die”, the Lord provided an abject lesson reminding Jonah of his mercy, compassion, forgiveness and concern for every living creature. God rebuked Jonah for his hardness of heart. The scripture do not tell us what happened to Jonah later.
Jonah’s actions demonstrated his ultimate submission to God’s directives, however the account does not record a specific confession or request for forgiveness.
The greater lesson is that Jonah’s three day stay in the oblivion of the great fish is a picture of Jesus’ three days in the tomb. On the third day, they were both delivered to achieve God’s eternal purposes!