Korea
From Memory Alpha, the free Star Trek reference
Korea was a peninsula of Asia. Located between the Yellow Sea on the west and the Sea of Japan on the east, the peninsula was separated from Japan by a narrow strait. The peninsula was home to two nations, and the northern nation shared a border with China. A major city of the southern nation was named Seoul.
This peninsula was identified on a political map of the North Pacific region. This map was accessed from the USS Enterprise library computer by the Talosians in 2254.(TOS: "The Cage")
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[edit] Background
Korea can be seen (barely visible) for a split second on a Pacific map flashed on the screen during "The Cage". Graphical enhancement of the image can make out the name "Korea" as well as a region which is intended to be the city of Seoul.
The term "Korea" can be applied to both the geographical region and the actual countries of North and South Korea. While the Korean peninsula was visible on the downloaded world map from "The Cage", the nation of Korea has never been mentioned in Star Trek.
Korea was also the site of the Korean War which has never been mentioned in Star Trek, although a Korean War service ribbon can be seen worn by a United States Air Force officer in TOS: "Tomorrow is Yesterday".

