Can the heating of the spring in oscillator be modeled by a velocity dependent force?

You can do it in an indirect way.

For example, consider a heating model where the energy being dissipated into heat at a given moment in time is proportional to the kinetic energy of the spring at that time, multiplied by some “dissipation constant”. In that way, the amount of energy going into heating the spring becomes implicitly dependent on the velocity of the spring thanks to $KE = frac{1}{2}mv^2$.

The standard way to model the internal energy dissipation (i.e. hysteretic damping, or in viscoelastic materials) for oscillatory motion is to make the stiffness term a complex number. The imaginary part of the number represents the energy dissipation.

The energy dissipation depends on the amplitude of vibration, but unlike viscous damping, if the amplitude is constant the same amount of energy is dissipated in each cycle of the vibration, independent of the frequency. This is consistent with experiments.