Yes, a quarterback can run the ball in football. Running the ball means moving the ball down the field in an attempt to make a down or even a touchdown. There aren't any rules that prohibit a player from running the ball while on the field. Any player on the field can run the football, and the quarterback is no exception. However, It is rare for the quarterback to run the football and score a touchdown but this can occur in some games.
The quarterback takes extra precautions that other players don't always have to take. Running the ball isn't a strategy that is implemented very often as it risks the health of the quarterback, who is often the most important player on the team or field at any given moment. The quarterback only usually runs if he is in danger of being sacked or tackled behind the line of scrimmage, which would be a loss of a down and yardage. Sometimes teams with mobile quarterbacks will run 'designed runs' in which the quarterback may run the football on purpose.