The association between two event times is of scientific importance in various fields. The local association measures capture the dynamic pattern of association over time, and it is desirable to examine the degree to which local association depends on different characteristics of the population. In this work, we adopt a novel quantile-based local association measure, and propose a conditional quantile association regression model to allow covariate effects in the local association analysis for bivariate survival data. An estimating equation for the quantile association coefficients is constructed on the basis of the relationship between this quantile association measure and the conditional copula. The asymptotic properties for the resulting estimators are rigorously derived. To avoid estimating density functions, we extend the induced smoothing idea to our proposed estimators in obtaining the covariance matrix. The proposed estimators and inference procedure are evaluated through simulations, and applied to an age-related macular degeneration (AMD) dataset, where we explore the association between AMD progression times in the two eyes of the same patient.