Summary: We consider the Toda system \[ \Delta u_i + \sum _{j = 1}^2 a_{ij}e^{u_j} = 4\pi \gamma _{i}\delta _{0}\text{ in }\mathbb{R}^2, \quad \int _{\mathbb{R}^2}e^{u_i} dx -1\), \( \delta _0\) is the Dirac measure at 0, and the coefficients \( a_{ij}\) are of the Cartan matrix of rank 2: \( A_2, B_2(=C_2),G_2\). Previously, the authors have gotten the classification and non-degeneracy results of solutions for Cartan matrix \( A_2\) and \( B_2\). In this paper, we consider the \( G_2\) case, and we completely classify the solutions and obtain the quantization result as well as the non-degeneracy of solutions for the \( G_2\) Toda system.