Audio-Visual Question Answering (AVQA) aims to answer textual questions by perceiving, understanding and integrating audio-visual information, which requires not only question-aware multimodal reasoning, but also precise spatial-temporal modeling to capture subtle dynamics. However, existing methods often rely on discrete, coarse-grained Top-K frame selection, ignoring temporal continuity. Some of them introduce question semantics in the early stage to enhance detail focus, but their guiding effect diminishes during reasoning. Although recent graph networks improve fine-grained modeling, the static connections still constrain dynamic adaptation to specific questions. To address these limitations, we propose QSTP, a framework that constructs patch-level spatial-temporal graphs, dynamically adjusts neighbor patch weights via question-aware aggregation, and continuously attends to information highly relevant to the question. We design an audio-driven visual gating module to highlight audio-related regions for better audio source localization, and a question-guided fusion strategy is put forward to further refine cross-modal alignment before answer prediction. Extensive experiments on the MUSIC-AVQA and MUSIC-AVQA-R datasets demonstrate that QSTP achieves state-of-the-art performance across various complex reasoning tasks.