This paper reports the spatiotemporal dynamics of an enzyme-catalyzed system with no-flux boundary conditions. We demonstrate that the diffusion-free system has a Hopf bifurcation, generating stable periodic solutions in the supercritical case and unstable ones in the subcritical case. For reaction-diffusion system, we explore and address the instability issue of the periodic solution in the Turing sense by employing the regular perturbation method. Our analysis reveals that specific combinations of diffusion coefficients can induce Turing instability in periodic solutions. Ultimately, numerical simulations validate these theoretical results and visually illustrate the spatiotemporal oscillatory behavior of the system.