There are no traditional Greek girl names that start with Q. The Greek language has no letter Q in its modern alphabet, so no authentic Greek name begins with it. Oddly, the letter itself has Greek roots. The Latin Q comes from an old Greek letter called koppa, which the Greeks borrowed from the Phoenicians.
The reason for the gap is simple. Koppa stopped being used in everyday Greek writing long ago, and today it survives only as a symbol for the number 90. Modern Greek has 24 letters, and none of them is Q. When Greek names are written in English, they are spelled with the Latin alphabet, and the “k” sound is almost always written with K or C. That is why Greek names like Kassandra, Katerina, and Kalliope start with K, and why none of them start with Q.
If you like the look or sound of a Q name, there are close options. Names such as Kyra, Kassia, and Calliope have the same hard “k” sound that Q often stands for in English. Some parents also choose Latin names such as Quintina and give them a Greek style, but these are not Greek in origin. For a true Greek girl name, the best choices begin with K, C, or another letter of the Greek alphabet.