In this paper, we consider the long-time dynamical behavior of the MGT-Fourier system

$$\{\begin{array}{l}{u}_{ttt}+\alpha u_{tt}-\beta \mathrm{\Delta }{u}_t-\gamma \mathrm{\Delta }u+\eta \mathrm{\Delta }\theta +f_1(u,u_t,\theta )=0,\\ {\theta }_t-\kappa \mathrm{\Delta }\theta -\eta \mathrm{\Delta }{u}_{tt}-\eta \alpha \mathrm{\Delta }{u}_t+f_2(u,u_t,\theta )=0.\end{array}$$

First we use the nonlinear semigroup theory to prove the well-posedness of the solutions. Then we establish the existence of smooth finite dimensional global attractors in the system by showing that the solution semigroup is gradient and quasi-stable. Furthermore, we investigate the existence of generalized exponential attractors.