Considering the interaction between two-mode SU(1,1) coherence states and a two-level atom, we find an exact solution of the Milburn equation for the system. We investigate the time evolution of the entanglement of field-atom and one of two modes of field using the quantum-reduced entropy and quantum relative entropy, respectively. The influences of intrinsic decoherence and two-mode photon number difference on the evolution of the entanglement of field-atom and one of two modes of field are discussed. It is shown that the entanglement of field-atom reduces to a stationary value with time evolution and the entanglement between two modes increases to another stationary value with intrinsic decoherence, and the values only depend on the two-mode photon number and mean photon number.