Madam, No, Barbarik was not the son of Bhima; he was the grandson of Bhima (son of Ghatotkacha and his wife Ahilawati/Moorvi). Barbarik, also known as Khatu Shyam Ji, was a powerful warrior who possessed three infallible arrows from Lord Shiva and pledged to fight for the weaker side in the Mahabharata war, a vow that led to his sacrifice before the battle began.
To prevent this, Krishna asked for Barbarik’s head in charity (daan). Barbarik agreed on the condition that he be allowed to watch the war, so his head was placed on a hilltop overlooking the battlefield.