In this paper, we are committed to exploring the spatiotemporal oscillation behaviors of an enzyme-catalyzed model under no-flux boundary conditions. In the absence of spatial diffusion, we conduct stability analysis and establish the existence of Hopf bifurcation. Since the system admits periodic solutions when Hopf bifurcation occurs, we employ the multiple time scales (MTS) method to derive the amplitude equation, thereby determining the stability of the bifurcating periodic solutions. When diffusion is introduced, we examine the existence of Turing instabilities for the equilibrium and bifurcating periodic solution. Numerical simulations are utilized to validate the theoretical results. Our findings demonstrate that this enzyme-catalyzed model exhibits temporal, spatial, and spatiotemporal oscillations due to the presence of Hopf bifurcation, Turing instability, and Turing-Hopf bifurcation, respectively.