A quasi-analytical approach is developed for detecting period-doubling bifurcation emerging near a Hopf bifurcation point. The new algorithm employs higher-order Harmonic Balance Approximations (HBAs) to compute the monodromy matrix, useful for the study of limit cycle bifurcations. Prediction of the period-doubling bifurcation is accomplished very accurately by using this algorithm, along with a detailed approximation error analysis, without using numerical integration of the dynamical system. An example is given to illustrate the results.