What will be the force exerted on the cue stick by the cue ball?

Force is equal to Mass times Acceleration, or F=MA. The mass is constant at 166 grams (for standard American pool balls). Acceleration is the square of centimeters per second (cm/s^2).

To properly answer your question, you would need to know how far the object ball moved and how long it took to move that distance. Then you can determine the Force.