Summary: The discrete Fourier analysis on the \(30^\circ\)-\(60^\circ\)-\(90^\circ \) triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group \(G_{2}\), which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of \(m\)-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.