Quantum walks are the quantum mechanical analog of classical random walks and an extremely powerful tool in quantum simulations, quantum search algorithms, and even for universal quantum computing. In our work, we have designed and fabricated an 8-by-8 two-dimensional square superconducting qubit array composed of 62 functional qubits. We used this device to demonstrate high-fidelity single- and two-particle quantum walks. Furthermore, with the high programmability of the quantum processor, we implemented a Mach-Zehnder interferometer where the quantum walker coherently traverses in two paths before interfering and exiting. By tuning the disorders on the evolution paths, we observed interference fringes with single and double walkers. Our work is a milestone in the field, bringing future larger-scale quantum applications closer to realization for noisy intermediate-scale quantum processors.