Our work develops a regularized tensor quantile regression framework for scalar responses and enables the robust analysis of tensor-variate data. Beyond our presented application to neuroimaging for identifying regions of the hippocampus implicated in Alzheimer's disease, tensor-variate data are ubiquitous in medicine, ecology, and other fields where large volumes of data are collected. The tensor quantile model considered in our work stands separate from previously-established tensor regression frameworks and requires its own theoretical investigation. We establish important statistical properties of our tensor effect estimator and convergence properties of our proposed estimation algorithm. To address the high dimensionality of the tensor quantile model and the non-differentiability of the quantile loss function, we assume that the tensor effect admits a Tucker decomposition and perform estimation using smoothing techniques combined with a block relaxation algorithm. Unlike previous two-stage approaches, our methodology simultaneously considers tensor decomposition and model estimation, ensuring that the decomposition is optimally-predictive of the response.