The longitudinal individual response profiles could exhibit a mixture of two or more phases of increase or decrease in trend throughout the follow up period, with one or more unknown transition points usually referred to as breakpoints or change points. The existence of such unknown point disturbs the sample characteristics, so the detection and estimation of these points is crucial. Most of the proposed statistical methods in literature, for detecting and estimating change points, assume distributional assumption that may not hold. A good alternative in this case is to use a robust approach which is the quantile regression model. There are trials in the literature to deal with quantile regression models with a change point. These trials ignore the within subject dependence of longitudinal data. In this paper we propose a mixed effect quantile regression model with a change point to account for dependence structure in the longitudinal data. Fixed effect parameters, in addition to the location of the change point, are estimated using profile estimation method. The stochastic approximation EM algorithm is proposed to estimate the fixed effect parameters exploiting the link between asymptotic Laplace distribution and the quantile regression. In addition, the location of the change point is estimated using the usual optimization methods. A simulation study shows that the proposed estimation and inferential procedures perform reasonably well in finite samples. The practical use of the proposed model is illustrated using a COVID-19 data. The data focus on the effect of global economic and health factors on the monthly death rate due to COVID-19 during from the 1st of April 2020 till the 31st of April 2021.