The Chinese were one of the civilizations visited by the alien being Kukulkan in the ancient past. The alien creature influenced Chinese architecture and agricultural development. Kukulkan told the Chinese to build a city, and when it was finished he would return. The Chinese only incorporated some of the designs into their architecture, such as the gateways and many dragon symbols, and never completed the energy amplification system that was supposed to signal Kukulkan. Kukulkan faded into mythology and was remembered as the Chinese dragon. (TAS: "How Sharper Than a Serpent's Tooth")
After having traveled through the Guardian of Forever to 1930 New York City, Spock and James T. Kirk were caught by a NYPD officer stealing clothes off a clothesline. When Kirk observed that the officer had noticed Spock's unusual appearance, Kirk attempted explained that "my friend is obviously Chinese." During a further attempt to explain Spock's ears, Spock reminded Kirk of "the unfortunate accident I had as a child," to which Kirk explained to the officer, that Spock "caught his head in a mechanical rice picker. But fortunately, there was an American missionary living close by who was actually a, skilled plastic surgeon in civilian life." The officer did not buy the story, and Kirk and Spock took off running with their stolen clothes. (TOS: "The City on the Edge of Forever")
In 2371 Harry Kim, asked by Captain Janeway about Neelix's food, recited the ancient Chinese curse "May you live in interesting times." (VOY: "The Cloud") The following year, when Tom Paris was sharing his feelings about Kes, Kim recited "an old Chinese expression" which stated, "stay out of harm's way." In response, Paris told Kim that "that's not a Chinese expression," to which Kim replied, "if it works, use it." (VOY: "Parturition")
Individuals of Chinese descent
The writer's guide for Star Trek: Voyager stated that Harry Kim (like actor Garrett Wang who played him) was of Chinese ancestry, which may be the source for his mentions of Chinese proverbs, but he himself was not an inhabitant of China – it was later revealed that his family resided in South Carolina.
It should be noted that the name "Kim" is common amongst Koreans.