His prayer is found in Jonah 2:2-10. We are not sure whether it is what he actually said in the belly of the fish or a psalm of thanksgiving when he emerged.
Yes, the Hebrew verbs are in the past tense, but Biblical tenses are rather fluid.
The genuineness of Jonah’s cry is patent, though his poetical capacity is unimpressive. He echoes the Book of Psalms, not that he is quoting but merely using conventional phraseology. This is my translation:
Jonah prayed to the Lord his God from the belly of the fish. He said:
Out of my affliction did I cry to the Lord, and He answered me.
From deep in the grave I cried: You heard my voice.
For You cast me into the deep, into the midst of the seas;
The flowing waters surrounded me:
All Your waves and billows passed over me.
I said: I am cast out from Your presence:
Yet will I once again gaze on Your holy temple.
The waters surrounded me to my very soul,
The deep encompassed me: the reeds enwrapped my head.
I went down to the bottoms of the mountains,
The earth was barred against me for ever –
But You brought me up alive from the grave, O Lord my God.
When my soul fainted within me, I remembered the Lord;
My prayer came before You, into Your holy temple.
They that serve false gods forsake their own good:
But I shall sacrifice to You with the sound of thanksgiving;
What I have vowed I shall pay:
Deliverance is from the Lord.