The method of constructing quantum error-correcting code by the quaternary self-orthogonal sub-code chains and L-chains are investigated in this paper. Random searching method is used to find quaternary self-orthogonal codes and these self-orthogonal codes could form some self-orthogonal code chains. By using combinatorial method, quantum error-correcting codes are constructed in light of L-chains which are constructed by dual of these self-orthogonal codes. The code chains of these self-orthogonal sub-codes and the L-chains which are obtained from the dual of these self-orthogonal sub-codes of length n between 20 and 36 and n=40,45,50,55,60 and dual distance five and six are determined. Some quantum codes of distance five and six are constructed by the obtained L-chains, some quantum error-correcting codes are new.