The letter q isn't traditionally used for transliterating the Greek alphabet to the Latin alphabet. As far as I know, it can be used for either representing the pre-classical letter Qoppa, or for informal, modern transliterations of the letter Theta, neither which is very useful for finding a deity or creature starting with the letter.
You might want to look for a deity or creature starting with a C in English instead, such as Calliope, Cronus or Cerberus.
There is one name in Roman Mythology, Quirinus, it's another name for the God of War, Ares in Greek Mythology
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