Both Green's theorem and Stokes' theorem involve the integral of a curl and it is easy to see that Green's theorem is a planar version of Stokes' theorem.
However, the divergence theorem involves the integral of a divergence, so how can it possibly be derived from Green's theorem?
Let me consider the $\mathbb{R}^2$ plane to be immersed in $\mathbb{R}^3$, so that i could use rotation and vector product, but everything I'll say can be said without this immersion as well.
For a vector field on a plane, $\vec{A} = (A_x, A_y, 0)$, you can define another vector field on a plane, $\vec{A^\times} = \vec{e}_z \times \vec{A} = (-A_y, A_x,0)$. Note that $ \vec{\nabla}\times \vec{A^\times} = (0,0,\vec\nabla\cdot\vec{A})$, and for $\vec{dl} = (dx, dy, 0)$ we have $(0,0,\vec{A^\times}\cdot \vec{dl}) = \vec{A} \times \vec{dl} $, so applying Green's theorem to $\vec{A^\times}$ you'll get the divergence theorem for $\vec{A}$.