Background information Edit
In 1964, Hong Kong was a British colony.
According to a 21st century-era political map of Earth, Hong Kong was a Special Administrative Region in southeastern China. It was located on the South China Sea coastline. This Central Intelligence Agency map, which was seen in the remastered "The Cage", was published either in April 2007 or September 2008.   The difference, which was not noticeable at the resolution of the image from the episode, was the addition of Kosovo, which gained its independence between the releases. The label "Hong Kong S.A.R." was not legible to the viewer.
It is possible that Hong Kong was on a 20th century-era political map of Earth used in the original "The Cage". However, due to the poor quality of the image and the inability to access the source material, this is far from certain.
Hong Kong was mentioned in an ultimately excised line of dialogue from the final draft script of TOS: "Miri", in which Uhura (whose part in the episode was eventually rewritten for John Farrell) commented that a newly discovered duplicate of Earth could detect a reply signal from the USS Enterprise "all the way to Hongkong... or whatever they call it down there."