An experimental study of the vibrational characteristics of a quartz tuning fork at low temperatures has been carried out. The measurement results in vacuum at 1.5 K are shown to almost coincide with data obtained at 0.7 K in the case of a tuning fork covered with a helium film. The dependence of the oscillation velocity of the tuning fork legs on the excitation force has been determined, and the dependence of the friction force on the oscillation velocity has been calculated. The results are analyzed together with the previously obtained data on the nonlinear dependence determined with the same tuning fork immersed into liquid helium during of turbulent fluid flow excitation. The oscillation velocity regions have been identified for which the internal friction effect in the tuning fork itself and the helium has a determining influence on tuning fork oscillations. The data have been analyzed using the nonlinear Duffing equation that enabled the determination of the temperature dependence of the tuning fork nonlinearity coefficient and the mutual friction coefficient caused by the turbulent flow of superfluid liquid.