Learning-based Behavior Graph Classification (BGC) is widely used in Internet infrastructure for partitioning and identifying similar behavior graphs, yet its real-world application faces notable challenges. The challenges are: (i) fine-grained emerging behavior graphs, and (ii) incremental model adaptations. To tackle these issues, we propose to (i) mine semantics in multi-source logs using Large Language Models (LLMs) under In-Context Learning (ICL), and (ii) bridge the gap between Out-Of-Distribution (OOD) detection and class-incremental graph learning. Based on these ideas, we develop the first unified framework termed as Fine-Grained and Class-Incremental Behavior Graph Classification (FG-CIBGC ). It consists of two novel modules, i.e., gPartition and gAdapt, that are used for partitioning fine-grained graphs and performing unknown class detection and adaptation, respectively. To validate FG-CIBGC, we introduce a new benchmark, including a 4,992-graph, 32-class dataset from 8 attack scenarios and a novel Edge Intersection over Union (EIoU) metric. Extensive experiments show FG-CIBGC outperforms baselines on fine-grained class-incremental BGC task and generates behavior graphs which enhance downstream tasks.