To mitigate the short cycle structure and poor error correction performance of the current quasi-cyclic low-density parity-check (QC-LDPC) codes, a novel construction method, based on the Fibonacci-Lucas sequence (FLS), is proposed for QC-LDPC codes with girth-8. In this construction method, the FLS is used to form an exponent matrix without girth-4, and then, a search algorithm identifies elements that eliminate girth-6 from the exponent matrix, yielding an incremental sequence. The corresponding exponent matrix is constructed to obtain its parity check matrix. Simulation results show that compared with three other QC-LDPC codes with the same code-rate and code-length, the constructed FLS-QC-LDPC code significantly improves the net coding gain at a bit error rate of 1×10⁻⁶, outperforming its counterparts in terms of error correction. In addition, the proposed construction method can flexibly select the code length and rate while maintaining a low computational complexity, which is efficient and adaptable.