We study the zeros of random power series with stationary complex Gaussian coefficients, whose spectral measure is absolutely continuous. We analyze the precise asymptotic behavior of the radial density of zeros near the boundary of the circle of convergence. The dependence of the coefficients generally reduces the density of zeros compared with that of the hyperbolic Gaussian analytic function (the i.i.d. coefficients case), where the spectral density and its zeros plays a crucial role in this reduction. We also show the relationship between the support of the spectral measure and the analytic continuation at the boundary of the circle of convergence.