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I would assume this is just a short form of Chiquita. Chiquita is from the Spanish for "very small" and has been used as a nickname, and occasionally an official given name, for women in Latin America for several generations. It doesn't surprise me that a woman of Hispanic ancestry in the USA named Chiquita would want to shorten it to Chiq today, because of course Chiquita has become so identified with the brand name of bananas in this country.