Accurate and automated lesion segmentation in Positron Emission Tomography / Computed Tomography (PET/CT) imaging is essential for cancer diagnosis and therapy planning. This paper presents a Swin Transformer UNet 3D (SwinUNet3D) framework for lesion segmentation in Fluorodeoxyglucose Positron Emission Tomography / Computed Tomography (FDG-PET/CT) scans. By combining shifted window self-attention with U-Net style skip connections, the model captures both global context and fine anatomical detail. We evaluate SwinUNet3D on the AutoPET III FDG dataset and compare it against a baseline 3D U-Net. Results show that SwinUNet3D achieves a Dice score of 0.88 and IoU of 0.78, surpassing 3D U-Net (Dice 0.48, IoU 0.32) while also delivering faster inference times. Qualitative analysis demonstrates improved detection of small and irregular lesions, reduced false positives, and more accurate PET/CT fusion. While the framework is currently limited to FDG scans and trained under modest GPU resources, it establishes a strong foundation for future multi-tracer, multi-center evaluations and benchmarking against other transformer-based architectures. Overall, SwinUNet3D represents an efficient and robust approach to PET/CT lesion segmentation, advancing the integration of transformer-based models into oncology imaging workflows.