Deep learning methods have recently achieved remarkable performance in vessel segmentation applications, yet require numerous labor-intensive labeled data. To alleviate the requirement of manual annotation, transfer learning methods can potentially be used to acquire the related knowledge of tubular structures from public large-scale labeled vessel datasets for target vessel segmentation in other anatomic sites of the human body. However, the cross-anatomy domain shift is a challenging task due to the formidable discrepancy among various vessel structures in different anatomies, resulting in the limited performance of transfer learning. Therefore, we propose a cross-anatomy transfer learning framework for 3D vessel segmentation, which first generates a pre-trained model on a public hepatic vessel dataset and then adaptively fine-tunes our target segmentation network initialized from the model for segmentation of other anatomic vessels. In the framework, the adaptive fine-tuning strategy is presented to dynamically decide on the frozen or fine-tuned filters of the target network for each input sample with a proxy network. Moreover, we develop a Gaussian-based signed distance map that explicitly encodes vessel-specific shape context. The prediction of the map is added as an auxiliary task in the segmentation network to capture geometry-aware knowledge in the fine-tuning. We demonstrate the effectiveness of our method through extensive experiments on two small-scale datasets of coronary artery and brain vessel. The results indicate the proposed method effectively overcomes the discrepancy of cross-anatomy domain shift to achieve accurate vessel segmentation for these two datasets.