Thermal infrared (TIR) object tracking often suffers from challenges such as target occlusion, motion blur, and background clutter, which significantly degrade the performance of trackers. To address these issues, this paper proposes a novel Siamese Motion Mamba Tracker (SMMT), which integrates a bidirectional state-space model and a self-attention mechanism. Specifically, we introduce the Motion Mamba module into the Siamese architecture to extract motion features and recover overlooked edge details using bidirectional modeling and self-attention. We propose a Siamese parameter-sharing strategy that allows certain convolutional layers to share weights. This approach reduces computational redundancy while preserving strong feature representation. In addition, we design a motion edge-aware regression loss to improve tracking accuracy, especially for motion-blurred targets. Extensive experiments are conducted on four TIR tracking benchmarks, including LSOTB-TIR, PTB-TIR, VOT-TIR2015, and VOT-TIR 2017. The results show that SMMT achieves superior performance in TIR target tracking.