When analyzing bivariate outcome data, it is often of scientific interest to measure and estimate the association between the bivariate outcomes. In the presence of influential covariates, conditional association measures can quantify the strength of association without the disturbance of the marginal covariate effects, and provide cleaner and less-confounded insights into the bivariate association. In this work, we propose estimation and inferential procedures for assessing the conditional Kendall's tau coefficient, by adopting the quantile regression and quantile copula framework to handle marginal covariate effects. The proposed method can flexibly accommodate right censoring and be readily applied to bivariate survival data. It also facilitates an estimator of the conditional concordance measure, namely a conditional C-index, where the unconditional C-index is commonly used to assess the predictive capacity for survival outcomes. The proposed method is flexible and robust, and performed satisfactorily in extensive simulation studies with and without censoring. Application of our methods to two real-life data examples demonstrates their desirable practical utility.