As a long-standing research area, class incremental learning (CIL) aims to effectively learn a unified classifier along with the growth of the number of classes. Due to the small inter-class variances and large intra-class variances, fine-grained visual categorization (FGVC) as a challenging visual task has not attracted enough attention in CIL. Therefore, the localization of critical regions specialized for fine-grained object recognition plays a crucial role in FGVC. Additionally, it is important to learn fine-grained features from critical regions in fine-grained CIL for the recognition of new object classes. This paper designs a network architecture named two-branch attention learning network (TBAL-Net) for fine-grained CIL. TBAL-Net can localize critical regions and learn fine-grained feature representation by a lightweight attention module. An effective training framework is proposed for fine-grained CIL by integrating TBAL-Net into an effective CIL process. This framework is tested on three popular fine-grained object datasets, including CUB-200-2011, FGVC-Aircraft, and Stanford-Car. The comparative experimental results demonstrate that the proposed framework can achieve the state-of-the-art performance on the three fine-grained object datasets.