You're sort of right.

The Latin alphabet, which English uses, began in around the 7th Century BC. It was an adapted version of the Greek alphabet at the time. The Greek alphabet itself was based on the Phonecian alphabet.

The Phonecian language had a /k/ sound - technically an "voiceless velar oral stop", similar to the first sound in English cat and kettle. Phoenician, being a semitic language, also had a similar sound, that was pronounced either farther back in the mouth, or with a degree of pharyngeal constriction. This is called a "voiceless uvular oral stop", or a "voiceless pharyngealized velar oral stop", and is present in modern-day Arabic and Hebrew.

The first of these sounds was represented in Phonecian with the letter that came to be Latin K, while the second was represented with the one that came to be Q. Greek only has the /k/ sound, so they used both of these symbols interchangeably for the same thing. Eventually kappa became the dominant form, and koppa (the Q) dropped out of use.

Latin did the same for a while, using K and Q interchangeably (and C too, but that's another story for another day), but after a while they settled on some stability. In Latin, the /k/ sound was actually pronounced differently before rounded vowels, which are vowels pronounced with a rounding of the lips (in Latin, these vowels are /o/ and /u/). The /k/ sound was 'labialized', which means it was pronounced with rounded lips too. So to mark this different pronunciation, the writers of the time took to writing it with Q instead of K (or C). Due to the historical relatedness of Latin and the Germanic languages, there is some correspondence between the Germanic wh- words, and Latin words with q-, but there are many Latin q- words that are not markers of relationships.

The spellings stuck, and, after sound changes over the millennia, the QU combination was essentially being pronounced as /kw/. In Middle English, following the Norman conquest and in the face of increasing Romanization of English culture, the /kw/ sound in English was spelled with QU. Previously it had been CW, but it was changed to make English look more like Latin and Old French. That's why native Germanic words like "quickly" and "queen" are spelled with a QU. (In Old English, they were spelled cwice and cwene.) Then of course, many French and Latin words were loaned into English, retaining the original spelling, which is where we get equine and bouquet.

More recently, we've gone full circle, and are now using Q to represent the uvular stop sound (which most European languages do not have) of Arabic and Hebrew when we transliterate them into Latin. Hence Iraq and Qatar; in Arabic the K and Q represent different sounds. You need to be careful when talking, or you may insult someone if you say kalb (dog) instead of qalb (heart)! I say we've gone "full circle", because if you recall, the ancestor of Q was what the Phoenicians used to represent this same sound.

tl;dr: history, and linguistics.