This article reports the spatio-temporal dynamics of a diffusive enzyme-catalyzed system. When spatial diffusion is absent, we perform the stability analysis and explore the Hopf bifurcation of the system. We also determine the stability of the periodic solution resulting from the Hopf bifurcation. In what follows, for the diffusive enzyme-catalyzed system, the precise occurrence conditions of the Turing instability are given. Finally, some complex pattern phenomena of the system are observed. The main contributions of this paper are: (1) The types of positive equilibrium are classified by adjusting the range of control parameter. (2) The strict Turing instability domain is outlined, theoretically. (3) Some complex pattern phenomena of the enzyme-catalyzed reaction system are observed by utilizing the numerical approach.