Every word in this list is spelled qu-. Queen, quiet, quiz. That is the English rule, and this generator does not pretend otherwise.
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Old English wrote this sound as C or K. Norman scribes brought qu- after 1066 and it stuck. Queue, queen, quiet, quiz — all of them. The famous exceptions (qi, qat, qoph) came in later from other languages. They are not in this list. Q is also 10 points in Scrabble, which is why people care about those exceptions at all.
No. Every word here is spelled qu-. Qi and qat are legal in Scrabble when you have no U, but they are not in this generator.