This study introduces a novel quadruped robot, the TerraAdapt, furnished with an innovative deformable wheel-foot integrated structure. This unique design grants the robot the flexibility to alternate between wheeled and footed modes of locomotion, making it efficient in traversing diverse terrains, from smooth indoor floors to challenging outdoor landscapes laden with obstacles. The study delineates an in-depth design and analysis of the deformable wheel and its integrated wheel-foot structure using screw theory. We engineer a 2RRR-RP wheel-foot mode-switching mechanism by modifying a 2RRR spatial six-bar mechanism with an additional RP branch. This mechanism aids in seamless transitioning between different movement modes. Moreover, a 2RRR parallel structure is employed to construct the footed mode structure.To substantiate the viability and efficacy of the proposed design, we carry out extensive motion simulations and construct an experimental prototype for field testing. The field trials reveal the robot's adeptness in adapting to varied terrains, highlighting the possible advantages of incorporating the proposed deformable wheel into micro mobile robot designs.