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I remember when the Muslim holy book was the Koran when I was in middle school, but now it's the Quran. But it's always been Qatar and Iraq (but still Kuwait.)
Who decided that 'Q' was going to be represent that sound instead of 'K', and why?

In Arabic, in fact, they've always been separate sounds! The sound we write "k" is spelled with the letter ك in Arabic, and is pronounced a little bit further forward in the mouth; the sound we write "q" is spelled with the letter ق and pronounced a little bit farther back. In phonetic terms, "k" is a velar sound, and "q" is a uvular sound.

English doesn't distinguish between these two different sounds. But since we've got a spare letter lying around—the English letter "q" is actually related to the Arabic letter ق, if you go back far enough!—it's become conventional to separate them in writing. "Kuwait" is spelled with a ك, so it gets a "k"; "Qatar" is spelled with a ق, so it gets a "q".

Nowadays, by the way, the letter Q has started to be associated with Arabic in particular, since it shows up significantly more there than in English. So in pop culture, you'll sometimes see names like "Hakeem" respelled to "Haqim"—even though the original Arabic has a ك in it! Linguistically, this is called hypercorrection, and it's the same reason you'll see words like "forté" and "afficioñado": English-speakers associate é with Romance languages and ñ with Spanish, but they have no special meaning in English, so they sometimes get inserted where they aren't actually needed.

I was going to propose Julius Klaproth, in his 1823 book Asia Polyglotta. He notates the difference between ك and ق as k versus q. In earlier works such as Hamer 1806 Ancient alphabets
both were represented as "k" with a note that [q] ق is "hard".
However, I see that
Christian Ravis 1649 in A discourse of the orientall tongues : viz. Ebrew, Samaritan, Calde, Syriac, Arabic, and Ethiopic together with a generall grammer for the said tongues p. 96 or 97 (in the "General grammar" part) notates the kaf / qaf distinction with k versus q as well.
Subsequently, it emerged that Otho Gualtperius 1590 in Grammatica linguae sanctae described the analogous Hebrew kaph / qoph distinction as kaph c,ch (=[k,x]) versus qoph "q or k".

So at this point I would say that it is unknown who first devised that convention.