Is is possible to introduce instrumental variables in a negative binomial regression i SPSS. If so, how is this done?

Not directly, but you could estimate the reduced form and then use the predicted values from that in a second stage. However, you would need to correct the standard errors from the second stage.

Thanks! That is certainly something I could do. Do you have any recommendations on what method to use to correct the standard errors in the second stage?

If this were an OLS regression, you would just need to recalculate the error variance estimate using the actual second stage variables and modify the coefficient statistics accordingly. It's a little more complicated. Look at the estimation details in the Algorithms doc to figure out what to modify. Offhand, I don't know.