We introduce an asymptotic expansion of the transition density of a nonlinear oscillator involving a jump-diffusion process. To calculate the asymptotic expansion, we first consider the composite of the solution for a nonlinear oscillator with a smooth function. If the jump-diffusion term is introduced with a small parameter, then the expectation of the composite can be asymptotically expanded with respect to that parameter. This leads to the asymptotic expansion of the density of the nonlinear oscillator. Then, we approximate the transition density by expanding the characteristic function of the solution of the nonlinear oscillator with respect to the small parameter, and present a numerical verification of the asymptotic expansion of the transition density.