How to open a Cygwin shell at a specific directory from Netbeans?

mintty can't directly call cd because that's a builtin command of the respective shell. What you really want is to start a shell in the correct directory.

I don't know about NetBeans or QuickOpener, but provided currentFolder contains an absolute Windows path, the following should work:

C:\cygwin\bin\mintty /bin/sh -lc 'cd "$(cygpath "$currentFolder")"; exec bash'

This runs a proper login shell that changes directory and then replaces itself with bash.

Not quite you asked for, but Cygwin comes with chere which sets up a context menu in Windows Explorer.

This may be a decent substitute for what you ask, and you might be able to check the mechanism that chere uses to fix your issue.