We propose an approach termed "qDAGx" for Bayesian quantile regression under directed acyclic graphs (DAGs) where these DAGs are individualized, in the sense that they depend on individual-specific covariates (x). A key distinguishing feature of the proposed approach is that the DAG structure is learned simultaneously with the model parameters, instead of being assumed fixed, as in most existing works. To scale up the proposed method to a large number of variables and covariates, we use for the model parameters the popular global-local horseshoe prior that affords a number of attractive theoretical and computational benefits to our approach. By modeling the conditional quantiles, qDAGx overcomes the common limitations of mean regression for DAGs, which can be sensitive to the choice of likelihood (e.g., an assumption of multivariate normality), as well as to the choice of priors. We demonstrate the performance of qDAGx via thorough numerical simulations and via an application in precision medicine by inferring person-specific protein--protein interaction networks in patients with lung cancer.