Answer: Origin of the Phrase 'Question Authority'

While I have not found a definitive answer to your question of who coined the phrase 'question authority’, here are some useful links to help you. Generally, the phrase has been attributed to the baby boomer generation, Timothy Leary, and a bumper sticker summation of Socrates' teachings.
This article: provides detailed history of anarchist/antiauthoritarian trends in the Unites States (try using control+F and typing in authority).
For a lecture on Socrates, try this link (again, using control+F when you get there):
And, for more information than you probably want on Timothy Leary, try this wikipedia link: