"Sheikh Mohammed bears full responsibility for anything that happens to me."
The video could not be immediately authenticated, while Emirati officials were not immediately available for comment.
Saudi Arabia, the UAE, Egypt and Bahrain severed diplomatic and trade ties with Qatar in June over allegations Doha supported Islamist extremists and had close ties to regional rival Iran. Doha denies the accusations.
In response to the video, Qatar's Foreign Ministry spokeswoman, Lulwa al-Khater, said in a statement that Doha was monitoring the situation.
"The state of Qatar observes the situation closely, and due to the total severance of ties with the UAE it is difficult to clearly establish the circumstances surrounding the situation," she said.
"Despite that, the State of Qatar principally stands with the protection of rights for every individual and affirms the entitlement of his family to pursue all legal means for the protection of his rights."