This paper presents a numerical approach for the determination of scattering frequencies of two-dimensional acoustic exterior resonance problems using boundary element method combined with a contour integral method. Since the solving range is surrounded by a Jordan curve in the complex plane, the complex eigenvalues can be acquired through evaluating the contour integral along the Jordan curve using the contour integral method. Complex scattering frequencies are directly extracted by the proposed approach, and their corresponding eigenmodes are also presented by evaluating the vibration amplitudes of internal points. The behaviors of the fictitious eigenfrequencies for unbounded domain are also studied. Numerical examples demonstrate the effectiveness of the approach for the resonance problems of unbounded domains.