For instance in words Iraq and Qashqai? Are there any historical reasons for that?
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There are a small number of words and names used in English that have Q but no U and do not correspond to a "kw" sound:
... among others
Most of these have a q because they are words or names that come from Arabic, which traditionally uses the letter Q to transcribe a sound that doesn't occur in English: the voiceless uvular stop, which sounds similar to but not quite like the sound of the letter "K", and is the sound usually used when saying these words in English.
A few of these—those starting with "QI", such as "Qi", "Qian", and "Qing"—come from Chinese, where Q is the letter used to transcribe a different sound—the voiceless alveolo-palatal affricate—which sounds similar to but not quite like the sound of the "CH", and is the sound usually used when saying these words in English.
The place names that you mention are transliterations from Arabic, and the letter Q is traditionally used in transliterations of Arabic to represent a stop sound which doesn't exist in English.
I remember from someplace that [older] Q words in English come from Latin words of Etruscan origin, and are always followed by a U. I guess that would probably apply to Romance languages too, but if so, then how come the French say "cinq" instead of "cinque"?