Quadruped robots have been developed for various purposes including surveillance, transportation and the other service applications. However, the locomotion of a quadruped robot unavoidably accompanies translational and rotational vibrations at the robot body with multiple and varying frequency components. The vertical vibrations particularly deteriorate the surveillance accuracy of a quadruped robot. Therefore, this paper proposes a vibration suppression system of a quadruped robot for the purpose of precise surveillance. Based on the vibration analysis of a quadruped robot, called Cheetaroid-I Carrier, the proposed vibration suppression system is designed with a two-link manipulator in a bi-articular structure. The mechanical parameters of the proposed system, i.e., the frame lengths and the gravity spring constants, are optimally designed considering the vibration characteristics of the actual robot system. In the proposed system, a sky-hook controller is utilized to reject the vibration of an end-effector. The vibration suppression performance of the propose system is verified in both the time domain and the frequency domain.