In this paper, we use the elementary and analytic methods to investigate the computational problem of quadratic character sums related to quaternary and quintuple symmetric polynomials modulo a prime $p$, respectively. By applying properties of Gauss sums and those of third-order characters modulo $p$, we establish some identities for such quadratic character sums in the cases of $(3,p-1)=1$ and $(3,p-1)=3$.