Yes, you can extend your Felgo application with some Qt C++ code for taking a screenshot of the application window. See here to learn more about integrating C++ and QML. You can e.g. create a C++ Component that allows to take a screenshot of the application window and use it from QML to take a screenshot.
Here’s also a StackOverflow topic about taking screenshots of a QML ApplicationWindow with C++ that might help you to implement the C++ part.