In a conversation with Kathryn Janeway, the Leonardo da Vinci hologram asked her how she came to the Americas, she replied that it was "a long story," but that it involved "a Portuguese ship, some Turkish pirates, [and] a couple of hurricanes." (VOY: "Concerning Flight")
During the 19th century, a ship commanded by British Royal Navy Captain Cray was hit by a typhoon in the Pacific Ocean. While the ship sustained heavy damage in the storm, Cray and his crew were able to go against all odds, and arrive back in London eight months later in the tattered remains of his ship. (VOY: "Year of Hell")
While camping on Archer IV in 2151, the landing party from the Enterprise encountered a storm front, with gusting winds approaching 80 kph. Charles Tucker, a Florida native, thought nothing of it, "compared to a hurricane coming up through the Florida Keys." (ENT: "Strange New World")
As hurricanes are cyclonic by nature, at the center of the circulation there can be found an area containing the lowest atmospheric pressure and relative calm known as the "eye".
In 2032, after becoming trapped inside a graviton ellipse aboard Ares IV, Earth astronaut John Kelly described the interior of the ellipse as being "very calm, like [he was] in the eye of a hurricane." (VOY: "One Small Step")
In 2365, William T. Riker compared being in the breathable zone, found on Theta VIII, that was surrounded by ammonia storms located less than a kilometer away, to being in the eye of a hurricane. (TNG: "The Royale")