This article has studied the finite-region stability and \(H_\infty \) filtering issue for two-dimensional (2-D) Markov jump systems (MJSs) composed of Fornasini-Marchesini model. We focus on the transient behavior of the system and ensure that the filtering error system is finite-region stable (FRS) with the defined \(H_\infty \) performance. Establishing special recursive formulas and selecting appropriate Lyapunov functions, we obtain sufficient conditions to satisfy the requirements for designing \(H_\infty \) filter. Finally, the correctness of the method proposed in this paper is verified by means of an example with respect to the Darboux equation.