This study examines the statistical process control chart used to detect a parameter shift with Poisson integer-valued GARCH (INGARCH) models and zero-inflated Poisson INGARCH models. INGARCH models have a conditional mean structure similar to GARCH models and are well known to be appropriate to analyzing count data that feature overdispersion. Special attention is paid in this study to conditional and general likelihood ratio-based (CLR and GLR) CUSUM charts and the score function-based CUSUM (SFCUSUM) chart. The performance of each of the proposed methods is evaluated through a simulation study, by calculating their average run length. Our findings show that the proposed methods perform adequately, and that the CLR chart outperforms the GLR chart when there is an increased shift of parameters. Moreover, the use of the SFCUSUM chart in particular is found to lead to a lower false alarm rate than the use of the CLR chart.