From quarter +‎ back, in reference to position in front of the halfback position.

quarterback (n.) also quarter-back, U.S. football position, 1876, from quarter (n.1) + back (n.); so called from his position on the field at the start of play, between the halfback and the center. As a verb, "to play quarterback," by 1945.

So … why “quarter-back”? Because, in the most basic formation drawn up by Camp, this player was directly behind the “holder of the ball” (or “center”) and halfway behind the “half-back,” who himself was halfway between the quarterback and fullback.

As one might guess, the name did, in fact, have more than a little to do with the player’s starting position relative to the line of scrimmage. In the early iterations of the game, with the standard offensive formation, the quarterback was halfway between the holder of the ball (center) and halfback.

The phrase to quarterback, or quarterbacking, means to take control of something. This is a reference to the quarterback position in football, often considered the most important position in the game. A quarterback must know all of the ins and outs of both football teams playing and the rules of the game.