Mass spectrometry proteomics, characterized by spiky, spatially heterogeneous functional data, can be used to identify potential cancer biomarkers. Existing mass spectrometry analyses utilize mean regression to detect spectral regions that are differentially expressed across groups. However, given the inter-patient heterogeneity that is a key hallmark of cancer, many biomarkers are present at aberrant levels for a subset of, not all, cancer samples. Differences in these biomarkers can easily be missed by mean regression, but might be more easily detected by quantile-based approaches. Thus, we propose a unified Bayesian framework to perform quantile regression on functional responses. Our approach utilizes an asymmetric Laplace likelihood, represents the functional coefficients with basis representations which enable borrowing of strength from nearby locations, and places a global-local shrinkage prior on the basis coefficients to achieve adaptive regularization. Different types of basis transform and continuous shrinkage priors can be used in our framework. We apply this model to identify proteomic biomarkers of pancreatic cancer missed by previous mean-regression based approaches.