As Federico explained, systems of quadratic are as general as arbitrary systems of polynomial equations of any degrees. So a general method for solving such systems analytically is out of the question. It is only for special systems of quadratic equations where there is additional algebraic structure that one can solve the system explicitly.
A notable example is the classification of simple Lie algebras. It can be seen as solving for structure constants $f_{ij,k}$ which give the coordinate expression of the Lie bracket $[e_i,e_j]=\sum_k f_{ij,k} e_k$ where the $(e_i)$ is some basis of the Lie algebra. The quadratic equations are given by imposing the Jacobi identity. Something similar is done in conformal quantum field theory where the $f_{ij,k}$ become OPE coefficients and the quadratic equations are given by the so-called crossing equation, a kind of associativity statement.
Another remarkable example of quadratic system that can be solved is the one considered in my article with Chipalkatti "Quadratic involutions on binary forms" where we find all $2^s$ solutions predicted by Bezout's Theorem. The system is given on pages 6-7 in terms of rather complicated coefficients $\omega$ defined on page 35. The complete set of solutions of the system, the object of Theorem 3.2, is given explicitly on pages 8-9 of the arXiv version.