The quadruped robot has the advantage of having a high degree of freedom of motion. Therefore, it has been used in various robot applications such as exploration. However, the quadruped robot is not easy to control due to the large number of joints. Existing control methodologies using proportional-integral-differential (PID) or model predictive control show decent performance on uniform terrain. However, the control performance is degraded in the extreme terrain where significant non-linear disturbance occurs. To tackle this problem, we propose FLONIC-Q: fuzzy logic and neural network-based intelligent control system for quadruped robots on extreme terrain. The proposed system includes high-level and low-level control using fuzzy logic and machine learning to deal with disturbances in extreme terrain. As demonstrated in the experiments, the proposed method performs superior to existing system response methods regarding absolute trajectory error and time analysis in various trajectory types.