Learning predictive models from functional magnetic resonance imaging (fMRI) connectivity data is important for analyzing cognitive processes and neurological disorders. Most existing approaches adopt a single-task learning (STL) formulation, where models predict either cognitive task states or clinical diagnoses independently. This setting often limits generalization because related tasks share latent neural representations that remain unused. Multi-task learning (MTL) addresses this limitation by learning shared representations across tasks, enabling complementary information transfer and improved robustness. In this work, we propose the Brain Connectivity Transformer (BCT), a graph-based transformer architecture for joint cognitive task and disorder classification from fMRI connectivity graphs. BCT operates on region-of-interest (ROI) connectivity graphs and employs multi-head self-attention to model global interactions among brain regions. A dynamic ROI attention module identifies task- and disorder-relevant regions, while a weighted orthonormal clustering readout aggregates attention-refined features into cluster-level representations. Within the MTL framework, BCT jointly optimizes task and disorder objectives. Under subject independent evaluation, BCT achieves 85.13± 0.3% task accuracy and 95.30± 1.1% disorder accuracy, outperforming the strongest baseline by 4.51% and 7.08% , respectively. These results demonstrate the effectiveness of transformer-based MTL on connectivity graphs for joint cognitive task and disorder prediction. The source code is available at https://github.com/DDasChakladar/Brain-Connectivity-Transformer .