This paper reports results of the computation of the drag force exerted on an oscillating object in quantum turbulence in superfluid $^4$He. The drag force is calculated on the basis of numerical simulations of quantum turbulent flow about the object. The drag force is proportional to the square of the magnitude of the oscillation velocity, which is similar to that in classical turbulence at high Reynolds number. The drag coefficient is also calculated, and its value is found to be of the same order as that observed in previous experiments. The correspondence between quantum and classical turbulences is further clarified by examining the turbulence created by oscillating objects.