The period and its property of a Velocity-Dependent-Frequency nonlinear oscillator with cubic restoring force has not been discussed in literatures, what’s more, the traditional standard methods, i.e. perturbation methods or harmonic balance methods for determining approximations to the periodic solutions for this kind of oscillators may breakdown in their corresponding first-order calculations. In particular, the frequencies become singular for finite values of the amplitude. This paper firstly gives the exact period expression of this oscillator; based on the integrability condition of period integral expression the exact solution of special class of initial conditions for the oscillator is obtained by using harmonic balance method. The property of the period of the oscillator is studied, and the approximate analytical expression of the period is given by the complete elliptic integral of the first kind, and then the phenomenon and reason of somewhat like the square waves of the oscillation are analyzed. It shows that the period of the oscillator decays to zero finally when the amplitude increases to infinity, and the reason of square waves is the parameter of the system , the square waves will be squarer when the amplitude increases. At last, a Hermite interpolation method is presented for the periodic solution of the oscillator, the time variable is transformed into a new harmonically oscillating time of which the frequency is one-half the one of the oscillator, with the corresponding governing differential equation transforms into a form suitable for Hermite interpolation analysis. The solutions are compared with the numerical solution and results show good agreement.