Yes: model selection criteria, such as the BIC, the AIC, or the minimum length criterion, are commonly used in the literature to compare models based on their goodness of fit (and regularized for their complexity, ie for their number of free parameters).
Here, since the negative Binomial has 2 parameters (instead of only 1 for a Poisson distribution), it is going to be more penalized by the AIC and the BIC than the Poisson distribution.
However, the validity of these criteria rely on some strong assumptions, that you will need to verify and justify. For instance, using the BIC requires that your data are i.i.d., that you have enough of them, that you correctly obtained your Maximum Likelihood Estimators of the parameters of the models.
An interesting reference is Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociological methods & research, 33(2), 261-304.