An algorithm for solving quadratic, two-stage stochastic problems is developed. The algorithm is based on the framework of the Branch and Fix Coordination (BFC) method. These problems have continuous and binary variables in the first stage and only continuous variables in the second one. The objective function is quadratic and the constraints are linear. The nonanticipativity constraints are fulfilled by means of the twin node family strategy. On the basis of the BFC method for two-stage stochastic linear problems with binary variables in the first stage, an algorithm to solve these stochastic quadratic problems is designed. In order to gain computational efficiency, we use scenario clusters and propose to use either outer linear approximations or (if possible) perspective cuts. This algorithm is implemented in C++ with the help of the Cplex library to solve the quadratic subproblems. Numerical results are reported.