Yes, Taekwondo is Korean. Taekwondo was developed by a number of Korean martial artists during the twentieth century.
The martial art that would later be known as Taekwondo began in the 1930s and 1940s, when a number of Koreans, who had travelled to Japan to study, began training in Karate. In the 1940s and 1950s, they returned from Japan and opened martial arts schools in Korea.
In the 1950s and 1960s, there were efforts to unify the different styles of empty-handed martial arts that were being taught in Korea at the time. In April of 1955, a naming committee decided upon the name ‘Taekwondo’ for the martial art – a name chosen partly because of its similarity to ‘Taekkyeon’ – an ancient Korean martial art that Taekwondo had taken inspiration from.
The official language of Taekwondo is Korean, and there are a number of traditions in Taekwondo that are based on Korean cultural traditions.