The primary response variable of interest follows a generalized linear model with subject-specific random effects from multivariate mixed effects model for multiple longitudinal processes. The usual independence assumption of within-subject measurement errors for the longitudinal covariate processes may lead to considerable bias in the estimation of GLM regression parameters, when departures from this assumption exist. Further, it is interesting to investigate how the dispersion of the measurement errors may be affected by some explanatory variables. We propose a data-driven method for modeling the measurement error covariance structure that requires no distributional assumption on the random effects and provides parametric regressions without constraints. Asymptotic properties including consistency and normality are shown for regression parameters in both measurement error covariance and generalized linear model. The performance of the proposed method is illustrated through numerical studies.