This article introduces a novel shallow 3-D self-supervised tensor neural network in quantum formalism for volumetric segmentation of medical images with merits of obviating training and supervision. The proposed network is referred to as the 3-D quantum-inspired self-supervised tensor neural network (3-D-QNet). The underlying architecture of 3-D-QNet is composed of a trinity of volumetric layers, viz., input, intermediate, and output layers interconnected using an S -connected third-order neighborhood-based topology for voxelwise processing of 3-D medical image data, suitable for semantic segmentation. Each of the volumetric layers contains quantum neurons designated by qubits or quantum bits. The incorporation of tensor decomposition in quantum formalism leads to faster convergence of network operations to preclude the inherent slow convergence problems faced by the classical supervised and self-supervised networks. The segmented volumes are obtained once the network converges. The suggested 3-D-QNet is tailored and tested on the BRATS 2019 Brain MR image dataset and the Liver Tumor Segmentation Challenge (LiTS17) dataset extensively in our experiments. The 3-D-QNet has achieved promising dice similarity (DS) as compared with the time-intensive supervised convolutional neural network (CNN)-based models, such as 3-D-UNet, voxelwise residual network (VoxResNet), Dense-Res-Inception Net (DRINet), and 3-D-ESPNet, thereby showing a potential advantage of our self-supervised shallow network on facilitating semantic segmentation.