Quadruped robots manifest great potential to traverse rough terrains with payload. Numerous traditional control methods for legged dynamic locomotion are model-based and exhibit high sensitivity to model uncertainties and payload variations. Therefore, high-performance model parameter estimation becomes indispensable. However, the inertia parameters of payload are usually unknown and dynamically changing when the quadruped robot is deployed in versatile tasks. To address this problem, online identification of the inertia parameters and the Center of Mass (CoM) position of the payload for the quadruped robots draw an increasing interest. This study presents an adaptive controller based on the online payload identification for the high payload capacity (the ratio between payload and robot's self-weight) quadruped locomotion. We name it as Adaptive Controller for Quadruped Locomotion (ACQL), which consists of a recursive update law and a control law. ACQL estimates the external forces and torques induced by the payload online. The estimation is incorporated in inverse-dynamics-based Quadratic Programming (QP) to realize a trotting gait. As such, the tracking accuracy of the robot's CoM and orientation trajectories are improved. The proposed method, ACQL, is verified in a real quadruped robot platform. Experiments prove the estimation efficacy for the payload weighing from 20 $kg$ to 75 $kg$ and loaded at different locations of the robot's torso.