Yes, check out the Paris-Harrington theorem's use of indicators as one way to do this for a specific kind of Ramsey theorish combinatorial claim. It's kinda a tough slog but it's a really explicit construction of a model of PA which makes a true (in N) combinatorial claim false.
Generally speaking, however, there is no general constructive method to produce such models. However, I believe the indicator method from the Paris-Harrington theorem can be extended to work for cases where the issue is the provable totality of functions with a certain rate of growth.