To compare a probabilistic model f and an observed distribution e, you break up the data into levels. Then compute the chi-square statistic sum(f-e)^2/e. If there are two models, the statistic with the lesser value corresponds to the better model.

The main difference between the two models is how the variance depends on the mean. In the QP model, the variance is a multiple of the mean*, that is, a linear function. In the NB model, the variance is a quadratic function of the mean. If you have enough data, you may estimate the variance depending on the mean for some bins. If the relationship is linear, the QP is better. If the relationship looks quadratic, the NB model should be preferrable.

This paper should be useful:

https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1141&context=usdeptcommercepub

You should back-up that you have some resonable explanation for the overdispersion and that this fits to your observations. For instance, overdispersion can arise when there are more zeros then expected by the Poisson distribution. This is better explained by zero-inflated ro a hurdle model. Or there may be clusters in your data, and that should be accounted for by appropriate random effects.

*in the Poisson model variance and mean are equal (the factor is 1, and this factor is relased to be any positive real value in the QP model).