Generative Recommendation (GR) has emerged as a new paradigm in recommender systems. This approach relies on quantized representations to discretize item features, modeling users' historical interactions as sequences of discrete tokens. Based on these tokenized sequences, GR predicts the next item by employing next-token prediction methods. The challenges of GR lie in constructing high-quality semantic identifiers (IDs) that are hierarchically organized, minimally conflicting, and conducive to effective generative model training. However, current approaches remain limited in their ability to harness multimodal information and to capture the deep and intricate interactions among diverse modalities, both of which are essential for learning high-quality semantic IDs and for effectively training GR models. To address this, we propose Multi-Aspect Cross-modal quantization for generative Recommendation (MACRec), which introduces multimodal information and incorporates it into both semantic ID learning and generative model training from different aspects. Specifically, we first introduce cross-modal quantization during the ID learning process, which effectively reduces conflict rates and thus improves codebook usability through the complementary integration of multimodal information. In addition, to further enhance the generative ability of our GR model, we incorporate multi-aspect cross-modal alignments, including the implicit and explicit alignments. Finally, we conduct extensive experiments on three well-known recommendation datasets to demonstrate the effectiveness of our proposed method.