In previous work, we developed a sequential Bayesian approach to estimate conditional quantiles of a response variable–a problem framed as quantile regression [1]. The latent conditional quantile function was modeled via a sparse Gaussian process with inducing points. The asymmetric Laplace distribution was considered for the likelihood of the data. The inference of the posterior distribution over the inducing variables was recast as its Laplace approximation. The importance of inducing input locations to predictive accuracy was identified. As a result, we proposed a novel approach to optimally select their locations by leveraging the Gaussian process predictive variance. Moreover, we exploited uncertainties in the inducing variables to guide the acquisition of new observations of the response variable. Adaptive inducing point allocation and active learning were combined into a sequential algorithm, which further enabled automatic selection of the required number of inducing points. The present work adapts the above methodology, which focused on global predictive accuracy, to the problem of finding the optimum of the conditional quantile function–a problem framed as quantile optimization. New observations are selected using Thompson sampling by drawing samples from the posterior and choosing the corresponding optimal inputs. We leverage the decoupled strategy by [2] to efficiently sample from the sparse Gaussian quantile posterior within the Thompson sampling. This framework for quantile optimization is applied to the design of a hydrofoil under uncertainties. We focus on the hydrodynamic performance of the hydrofoil, with XFOIL as the flow solver, supplemented by analytical corrections to the hydrodynamic coefficients to account for free-surface proximity effects. Multiple sources of uncertainty are considered, including operational and manufacturing uncertainties. This application demonstrates the effectiveness of our algorithm and highlights its key advantages in high-dimensional uncertainty settings, as commonly encountered in engineering design [3].