In this paper, we obtain the necessary and sufficient conditions for quasi-cyclic codes with index even to be symplectic self-orthogonal. Then, we propose a method for constructing symplectic self-orthogonal quasi-cyclic codes, which allows arbitrary polynomials that divide $ x^{n}-1$ to construct symplectic self-orthogonal codes. Especially in the case of $1$-generator quasi-cyclic codes with index two, our construction improves Calderbank's additive construction, Theorem 14 in ``Quantum error correction via codes over $GF(4)$". Finally, we construct many binary symplectic self-orthogonal codes with excellent parameters to illustrate our approach's effectiveness. The corresponding quantum codes improve Grassl's table (bounds on the minimum distance of quantum codes. http://www.codetables.de).