A bridge-randomized penalization that employs a prior for the shrinkage parameter, as opposed to the conventional bridge penalization with a fixed penalty, often delivers more superior performance compared to many other traditional shrinkage methods. In this paper, we develop an efficient Bayesian computational algorithm via the two-block Markov Chain Monte Carlo method for the bridge-randomized penalization in quantile regression to perform inference in the high-dimensional "large-p" and "large-p-small-n" settings. To construct a fully Bayesian formulation, we utilize the asymmetric Laplace distribution as an auxiliary error distribution and the generalized Gaussian distribution prior for the regression coefficients. Simulation studies encompassing a wide range of scenarios indicate that the proposed method performs at least as well as, and often better than, other existing procedures in terms of both parameter estimation and variable selection. Finally, a real-data application is provided for illustrative purposes. © 2024 Informa UK Limited, trading as Taylor & Francis Group.