A power-law type stress relaxation leads to a fractional order viscoelasticity model. We consider the quasistatic fractional order viscoelasticity model which is mathematically expressed as a Volterra integral of the second kind with a weakly singular kernel. Employing the symmetric interior penalty discontinuous Galerkin (SIPG) method and linear interpolation in time, we derive the fully discrete problem. To mitigate temporal discretization errors from a physical perspective in the quasi-static state, we introduce an auxiliary discrete velocity using the Crank-Nicolson approximation. We present a priori stability and error analyses for both the semi-discrete and fully discrete schemes. Additionally, we provide a residual-based a posteriori energy error estimation. Finally, we conduct numerical experiments that validate the error estimate theorems and demonstrate their applicability using real material data.