State-of-the-art NAND flash chips experience significant channel noise impairments and raise several data reliability issues. To overcome these issues, low-density parity-check (LDPC) codes are becoming the main stream error-correctingcodes in flash memory controllers. Consequently, the long beliefpropagation (BP) decoding latency starts to deteriorate the system performance. In this paper, a low-complexity quantization-aware belief-propagation (QA-BP) decoding scheme is introduced for meeting the requirement of faster BP convergence speed. Using the shuffled BP algorithm, the proposed QA-BP scheme takes advantage of highly unequal error probability of input loglikelihood-ratios (LLRs), transpired as a result of non-uniform channel quantization, and update only less reliable variable nodes that are initiated with higher error probability LLR values. For realizing the QA-BP scheme, neither additional runtime operations nor real-valued comparisons are required. Using simulation results, it is shown that the proposed QA-BP scheme can yield a noticeable improvement in BP convergence speed without compromising the error performance.