The torque DOES increase as flowrate increases. But I think your assumption that fluid velocity exiting the impeller is wrong. At least the velocity of the fluid as it exits the pump volute (casing) certainly changes. This is evident on the face of it by looking at the fact that the flowrate has increased. Since the dimensions of the discharge piping is fixed, and the volumetric (and mass) flow rate is increased, then velocity must be higher. If the mass flow rate is higher, then the pump is doing more work. It is this higher work (per unit time) which requires a higher power input from the prime mover (let's assume an electric motor).
As a practical example of this, a longstanding good operating practice for motor driven pumps calls for the dicharge isolation valve to be closed (zero flow rate) at the time of starting the pump motor. This allows the motor to start with little load, and hence reduces the motor starting current surge. Otherwise sometimes the initial starting surge can trip the motor out at the start on an over-current trip. After the pump is up to speed, the discharge valve is opened to achieve the required flow.
Hope this helps.