This position in American football got its name from its location. The quarterback is the back whose position is one-quarter of the way between the line and the fullback.
It comes from the Latin verb quaerere, which means seek or ask. The past participle of the word, quaestus, formed the basis of a noun, quaestio. Quaestio evolved into the English word question.
While the word "draw" had been in use for some time in the American Old West, this expression didn't come about until 1903 when Cyrus T. Brady used it in his book, The Bishop. "He had the reputation...of being the quickest man on the draw...in the Territory."
Today's version of the expression is a short form of "quiet as a mouse in cheese," which came from The Pleasant History of Two Angry Women of Abington by Henry Porter. The line from the 1599 story went like this, "Mum, mouse in cheese, cat is neare." George Eliot produced today's version in his 1859 book, Adam Bede: "She looks as quiet as a mouse."
It's a Latin word whose literal meaning is "of who," but the way we use the word today originates from a Latin legal phrase. It was customary among English justices of the peace to name at least one person of especial knowledge or prudence without whose presence the other justices would be unwilling or unable to proceed with the business of the court. The wording of the naming of the justices contained the Latin phrase, "quorum vos Bob Smith, John Doe, and William Jones unum esse volumus, of whom we will that you, Bob Smith be one." The abbreviation "quorum" was applied to the specific persons who were required to be present before a session could be opened, and it ultimately developed into the meaning that is used today.