We consider the asymptotic behavior of the solution to a system of quadratic nonlinear Schrödinger equations in two dimensions. It is known that quadratic nonlinearities are critical in the sense of the asymptotic behavior of the solution. We show the existence of the solution to the final state problem by imposing an appropriate asymptotic behavior of the solution. In particular, we construct the asymptotic behavior of the solution by using the Jacobi elliptic functions.