Accurate liver vessel segmentation is essential for effective liver surgery pre-planning, and reducing surgical risks since it enables the precise localization and extensive assessment of complex vessel structures. Manual liver vessel segmentation is a time-intensive process reliant on operator expertise and skill. The complex, tree-like architecture of hepatic and portal veins, which are interwoven and anatomically variable, further complicates this challenge. This study addresses these challenges by proposing the UNETR (U-Net Transformers) architecture for the multi-class segmentation of portal and hepatic veins in liver CT images. UNETR leverages a transformer-based encoder to effectively capture long-range dependencies, overcoming the limitations of convolutional neural networks (CNNs) in handling complex anatomical structures. The proposed method was evaluated on contrast-enhanced CT images from the IRCAD as well as a locally dataset developed from a hospital. On the local dataset, the UNETR model achieved Dice coefficients of 49.71% for portal veins, 69.39% for hepatic veins, and 76.74% for overall vessel segmentation, while reaching Dice coefficients of 62.54% for vessel segmentation on the IRCAD dataset. These results highlight the method's effectiveness in identifying complex vessel structures across diverse datasets. These findings underscore the critical role of advanced architectures and precise annotations in improving segmentation accuracy. This work provides a foundation for future advancements in automated liver surgery pre-planning, with the potential to enhance clinical outcomes significantly. The implementation code is available on GitHub: https://github.com/saharsarkar/Multiclass-Vessel-Segmentation .