I'm not sure how to go about creating a solder mask. If I use the Fill Closed Boundaries option, then the pads would be completely covered.
How can I make a solder mask to match the green section in the attached pic?
You typically do not have to create the "Green Area". Typically the solder mask (on a gerber file) is "only" the enlarged pads (i.e. the Keep away area which appears "gray" on your image). The the PCB shop will essentially place solder mask everywhere else except where you have drawn the pads in your Gerber file. So creating the enlarged Pads and using FCB would be fine.
Here's what a typical solder mask gerber file looks like:
Please Let me know if that makes sense.