You may need to write a bit of C# code to accomplish your scenario.
We provide a sample project (check the Quet Web Parts installation folder and see the pleCode) that shows how to create your custom action and check the documentation about how to integrate it with QWP.
In your case you write the code to check in the document, build the assembly, place your assembly into the private bin (this is only temporary in order to have the custom action drop down control in ezEdit to be populated with your custom action code, after that you can move the assembly to the GAC), then create a custom action (see the docs). For this custom action the first operation would be your custom code that checks in the doc then the very next operation would be Save (this is out of the box functionality).
Please search this forum for "custom action" and you'll find more details about custom actions.
Any other questions please let me know.