When an object moves in a circle with a constant speed its velocity remains constant True or false?

No, it doesn't. Velocity is a vectorial quantity, it has magnitude (speed) and direction. Uniform circular motion can be given as an example in order to help you with your question. Even though the speed is constant in this particular example, the direction changes all the time.

The velocity of an object changes when the net force acting on it is not zero. When the net force is perpendicular to the velocity, it does no work and the kinetic energy of the object remains constant, so the force only changes the direction of velocity not the magnitude (speed).