Is the story of Hassan ibn Thabit's conversion to Islam true, after the disbelievers asked him to satirize the Prophet, may Allah bless him and grant him peace?

The story mentioned about Hassan ibn Thabit has not been found, nor has it been mentioned by his biographers. Rather, they state that he embraced Islam after the Hijra. However, his defense of the Prophet (peace be upon him) with his poetry and his satirical attacks on the disbelievers after his conversion to Islam are well-established.