In this paper, a novel benchmark for audio-visual question answering continual learning (AVQACL) is introduced, aiming to study fine-grained scene understanding and spatial-temporal reasoning in videos under a continual learning setting. To facilitate this multimodal continual leaning task, we create two audio-visual question answering continual learning datasets, named Split-AVQA and Split-MUSIC-AVQA based on the AVQA and MUSIC-AVQA datasets, respectively. The experimental results suggest that the model exhibits limited cognitive and reasoning abilities and experiences catastrophic forgetting when processing three modalities simultaneously in a continuous data stream. To address above challenges, we propose a novel continual learning method that incorporates question-guided cross-modal information fusion (QCIF) to focus on question-relevant details for improved feature representation and task-specific knowledge distillation with spatial-temporal feature constraints (TKD-STFC) to preserve the spatial-temporal reasoning knowledge acquired from previous dynamic scenarios. Furthermore, a question semantic consistency constraint (QSCC) is employed to ensure that the model maintains a consistent understanding of question semantics across tasks throughout the continual learning process. Extensive experimental results on Split-AVQA and Split-MUSIC-AVQA datasets illustrate that our method achieves state-of-the-art audio-visual question answering continual learning performance. Code is available at Supplementary Material.