In this paper, second-moments of the responses are analytically solved by the Laplace transform in a coupling two-degree-of-freedom system with a biexponentical dissipative memory kernel function driven by a thermal broadband noise. The mean square displacement x2(t)> is different from anomalous diffusion (i.e. x2(t)> ∝ tα (0αα≠1)), which is produced by the single-degree-of-freedom generalized Langevin equation. The oscillation-diffusion of x2(t)> with the change of time and noise parameters is observed generally. According to our analysis, a particle confined by the harmonic potential can escape with the help of the coupling-damping factor B. The diffusion of x2(t)> aggravates with B increasing. However, x2(t)> tends to the stationary state with the increase of the friction coefficient Further, if the two thermal noises are in cross-correlation, smaller cross-correlation time has a deeper influence on second-moments. Meanwhile, the diffusion aggravates and the cross-correlation between two displacements strengthens markedly with cross-correlation strength increasing. It is consistent with physical intuition.