Ohio is a region of the North American continent of Earth. This was a state in the nation known as the United States of America.
In 2372, when Quark, Rom, and Nog were transported back to 1947, Nog lied about an alien invasion that was going to attack Earth. He pointed to a random spot on a map, and Wainwright responded "They're invading...Cleveland." which is a major city in Ohio. (DS9: Little Green Men)
During a crisis in 2372, Captain Janeway and B'Elanna Torres consulted a study on the duplication of matter and antimatter preformed at Kent State University, which is located in Ohio. (VOY: Deadlock)
Chakotay said he had a cousin who lived in Ohio, to whom he wrote a letter in 2374, trying to use the Hirogen Communications Network to send it. (VOY: Message in a Bottle)
In an alternate timeline created by Vosk and the Na'kuhl, the Nazi/United States battle front went through Columbus, the capital of Ohio. (ENT: Storm Front, Part II)
The adventure film Mister Willis of Ohio featuring the title character Mister Willis takes place in this state. (ENT: "Cogenitor")