Longitudinal zero-inflated count data are widely encountered in many fields, while modeling the correlation between measurements for the same subject is more challenge due to the lack of suitable multivariate joint distributions. In this paper, we propose a novel approach by using copula in longitudinal zero-inflated regression model, solving both problems of specifying joint distribution and parsimoniously modeling correlations with no constraint. We then study the use of hyper-spherical coordinates to parametrize the correlation matrix in the copula in terms of a set of angles, effectively a new set of constraint-free parameters on their support. Aided by this property, we propose separated mean and correlation regression models to understand these key quantities, which can also handle irregularly and possibly subject-specific times points. We show that the resulting estimators of the proposed approaches are consistent and asymptotically normal. Data example and simulations support the effectiveness of the proposed approach.