We classify the quadratic extensions $K=\mathbb{Q}[\sqrt{d}]$ and the finite groups 𝐺 for which the group ring $\mathfrak{o}_K[G]$ of 𝐺 over the ring $\mathfrak{o}_K$ of integers of 𝐾 has the property that the group $\mathcal{U}_1(\mathfrak{o}_K[G])$ of units of augmentation 1 is hyperbolic. We also construct units in the $\mathbb{Z}$-order $\mathcal{H}(\mathfrak{o}_K)$ of the quaternion algebra $\mathcal{H}(K)=\left\frac{-1,-1}{k}(\right)$, when it is a division algebra.
