Structure of $x^2 + xy + y^2 = z^2$ integer quadratic form

Sure, you're just looking for generators of $O(f,\mathbb{Z})$, where $f(x,y,z)=x^2+xy+y^2-z^2$ is a quadratic form in 3 variables. Since $f$ is isotropic, this will be a non-uniform arithmetic fuchsian group, commensurable with $PSL_2(\mathbb{Z})$. One can start looking for generators by finding vectors $(x,y,z)\in \mathbb{Z}^3$ such that $f(x,y,z)=1, 2$. Reflections in such vectors will generate a reflective subgroup, which one can find by Vinberg's algorithm. If this subgroup is finite-index, then you'll have found something analogous to the Pythagorean triple generators. If not, then you'll have to look for some other generators. Usually, you can also include reflections in vectors of the form $f(x,y,z)=-1,-2$ as well, which give rotations of the hyperbolic plane $f(x,y,z)=-1$ up to $\pm 1$. If this group is reflective, then it will appear somewhere in Daniel Allcock's list.