Considering a double J-C model with intensity-dependent coupling, we have studied the effects of the intensity-dependent coupling, the mean photon numbers and the atomic motion, on the entanglement and quantum discord between the two two-level atoms when the moving atoms are initially in a maximally entangled state and the fields are in the single-mode thermal fields. The results show that, the entanglement and quantum discord disappear and revive periodically, and can have up to their starting values after revival. A rise in cavity temperature accelerates the death of the entanglement and quantum discord. In addition, the field-mode structural parameter has a strong effect on the entanglement and quantum discord in the system. When the field-mode structural parameter takes a suitable value, the entanglement and quantum discord of the two atoms can be kept from start to finish.