The"Q" signals were instituted by the late Ben Demby, a ham radio operator. He was the man who developed the City of Miam police communications system in the early 1940's. I knew Ben when I was in my teens and later his late son, Ben Jr., also an electronics man, for the Miami Herald.
ITU started pre-radio, it was the International Radiotelegraph Union. Yes "Q" signals began before radio. Other than the comments here I had no idea how they began to be used in Miami. I do recall that Miami was home to Pan American Airways and overseas flights used to always have a Morse operator for long distance communications and "Q" signals were often used in their Morse traffic. In fact there is a subset of "Q" signals specifically for aeronautical usage. So much for history.
another brief note. I looked at the 'wiki' link, the Morse table which shows footnote 8 from ITU R-M 1677-1 is incomplete. I was present in Geneve when this update to the Recommendation was adopted. The purpose was to ADD 'areobase' [@ ] as a new Morse character so that email addresses could be more easily transmitted, the new character is NOT shown in the wiki article although footnoe (8) is correct. Enough for the history!