This paper proposes a novel method to improve locomotion learning for a single quadruped robot using multi-agent deep reinforcement learning (MARL). Many existing methods use single-agent reinforcement learning for an individual robot or MARL for the cooperative task in multi-robot systems. Unlike existing methods, this paper proposes using MARL for the locomotion learning of a single quadruped robot. We develop a learning structure called Multi-Agent Reinforcement Learning for Single Quadruped Robot Locomotion (MASQ), considering each leg as an agent to explore the action space of the quadruped robot, sharing a global critic, and learning collaboratively. Experimental results indicate that MASQ not only speeds up learning convergence but also enhances robustness in real-world settings, suggesting that applying MASQ to single robots such as quadrupeds could surpass traditional single-robot reinforcement learning approaches. Our study provides insightful guidance on integrating MARL with single-robot locomotion learning.