Yes, the First Pitch Quarterback Football Machine and all First Pitch pitching machines can be used while they are still attached to the First Pitch Transporter Pro.
In fact, that is one of the main advantages of the Transporter Pro vs. the standard First Pitch Transporter or Transporter Jr. The handle can get in the way of pitches/throws with the regular transporters whereas the Transporter Pro lies flat on the ground where there is nothing obscuring the path of balls thrown from the machine.