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Researchers are often interested in understanding the relationship between a set of covariates and a set of response variables. To achieve this goal, the use of regression analysis, either linear or generalized linear models, is largely applied. However, such models only allow users to model one response variable at a time. Moreover, it is not possible to directly calculate from the regression model a correlation measure between the response variables. In this article, we employed the Multivariate Generalized Linear Mixed Models framework, which allows the specification of a set of response variables and calculates the correlation between them through a random effect structure that follows a multivariate normal distribution. We used the maximum likelihood estimation framework to estimate all model parameters using Laplace approximation to integrate out the random effects. The derivatives are provided by automatic differentiation. The outer maximization was made using a general-purpose algorithm such as \texttt{PORT} and \texttt{BFGS}. We delimited this problem by studying only count response variables with the following distributions: Poisson, negative binomial (NB) and COM-Poisson. The models were implemented on software \texttt{R} with package \texttt{TMB}. Besides the full specification, models with simpler structures in the covariance matrix were considered (fixed and common variance, fixed dispersion, $ρ$ set to 0). These models were applied to a dataset from the National Health and Nutrition Examination Survey, where three underdispersed response variables were measured at 1281 subjects. The COM-Poisson model full specified overcome the other two competitors considering three goodness-of-fit indexes. Therefore, the proposed model can deal with multivariate count responses and measures the correlation between them taking into account the effects of the covariates.