In this paper we establish the existence and uniqueness of global solutions (in time), as well as the existence, regularity and stability (upper semicontinuity) of the attractor for the semigroup generated by the solutions of a two-dimensional nonlinear hyperbolic-parabolic coupled system with fractional Laplacian. In addition, we also obtain the existence of an exponential attractor and show that this attractor has a finite fractal dimension in a space containing the phase space of the dynamical system.