Are there any in-built methods, or flags I can set, with a Java Graphics2D object such that I can draw polygons or other shapes with gaussian blur automatically applied?
Or if anyone knows of a computationally efficient shortcut/hack, that would be greatly appreciated.
AFAIK - no.
Most blur operations operate on a BufferedImage as they need to manipulate the underlying pixel data. A typical approach is to render the content to an image, blur it, then paint that to whatever you want
I prefer to use JHLabs filters generally, they're not the fastest, but generally give a high quality result.
This example is a little involved, but demonstrates the basic idea.
As another example and if your really want to make your eyes bleed ;)