Applications to Green's Theorem in Economics?
Here and here are two application of the theorem to finance.

Here is an application to game theory. 

This is an application of the theorem to complex Bayesian stuff (potentially useful in econometrics).

Also, the Green's theorem is used in this proof, which relates to dynamics system, and therefore could be related to economic models (source of this here)

Last but surely not least, this book might have relevant stuff. At least Chapter 12 does use it.