  We compute the integral Grothendieck rings of the moduli stacks,
$\mathcal{M}_2$, $\overline{\mathcal{M}}_2$ of smooth and stable curves of
genus two respectively. We compute $K_0(\mathcal{M}_2)$ by using the
presentation of $\mathcal{M}_2$ as a global quotient stack given by Vistoli. To
compute the Grothendieck ring $K_0(\overline{\mathcal{M}}_2)$ we decompose
$\overline{\mathcal{M}}_2$ as $\Delta_1$ and its complement
$\overline{\mathcal{M}}_2 \setminus \Delta_1$ and use their presentations as
quotient stacks given by Larson to compute their Grothendieck rings. We show
that they are torsion-free and this, together with the Riemann-Roch isomorphism
allows to ultimately give a presentation for the integral Grothendieck ring
$K_0(\overline{\mathcal{M}}_2)$.
