In this paper, a nondestructive method to measure bulk and shear viscosities of a homogeneous viscoelastic sphere using free vibration is investigated by taking into account a single Voigt model for viscous and elastic stress tensors. For the viscosity measurement, Q value characterizing the damping due to the viscous effect is used. The formulation of the Q value for spheroidal and torsional modes is theoretically made based on the redefinition of the coefficients of bulk and shear viscosities as positive values. Newly introduced coefficients A and B constituting Q value formula, which correspond to bulk and shear deformations in the free vibration, are numerically calculated for spheroidal and torsional modes, respectively. The method to measure the viscosity by using the observed values of vibration and coefficients A and B is presented, and the example to apply the method is also shown. This study may find applications in rheology of soft material where viscous effects of spherical structures matter.