Answer
No, Al-Husayn ibn al-Qasim did not belong to the Abbasid dynasty. The Abbasid dynasty ruled over much of the Middle East and beyond from the mid-8th to the 13th century. Al-Husayn ibn al-Qasim, on the other hand, was a member of the Zaydiyyah branch of Shia Islam and a descendant of Hasan ibn Ali, the grandson of the Prophet Muhammad.