This article deals with the problem of <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$l_{2}-l_{\infty }$ </tex-math></inline-formula> filtering for discrete-time Takagi–Sugeno fuzzy nonhomogeneous Markov jump systems with quantization effects, respectively. The time-varying transition probabilities are in a polytope set. To reduce conservativeness, a mode-dependent logarithmic quantizer is considered in this article. Based on the fuzzy-rule-dependent Lyapunov function, sufficient conditions are given such that the filtering error system is stochastically stable and has a prescribed <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$l_{2}-l_{\infty }$ </tex-math></inline-formula> performance index, respectively. Finally, a practical example is provided to illustrate the effectiveness of the proposed fuzzy filter design methods.