This study focuses on the locomotion capability improvement in a tendon-driven soft quadruped robot through an online adaptive learning approach. Leveraging the inverse kinematics model of the soft quadruped robot, we employ a central pattern generator to design a parametric gait pattern, and use Bayesian optimization (BO) to find the optimal parameters. Further, to address the challenges of modeling discrepancies, we implement a multi-fidelity BO approach, combining data from both simulation and physical experiments throughout training and optimization. This strategy enables the adaptive refinement of the gait pattern and ensures a smooth transition from simulation to real-world deployment for the controller. Moreover, we integrate a computational task off-loading architecture by edge computing, which reduces the onboard computational and memory overhead, to improve real-time control performance and facilitate an effective online learning process. The proposed approach successfully achieves optimal walking gait design for physical deployment with high efficiency, effectively addressing challenges related to the reality gap in soft robotics.