When you use the Paint Bucket tool in Inkscape it doesn't fill a "region". It creates a separate object with a fill. There's no such thing as a "region" in Inkscape. There are only objects, and objects can have fills or strokes.
For example, here I'm filling a triangle using the Paint Bucket, but you can see it's actually a separate object. You can select it and move it.
You can select one of those filled objects and apply a pattern fill in the Fill and Stroke panel.
Generally, it's not really a good idea to use the Paint Bucket to fill things. It's much better to select objects and apply a fill or pattern directly. All you are doing is creating more shapes to fill which is inefficient.
For your graphic, it would be much better to create triangles, apply fills directly to the triangles in the Fill and Stroke panel, and fill the circles white and overlay these over the vertices of the triangles. Also this will avoid those nasty white gaps the Paint Bucket tool can also create if you don't adjust the settings properly (as can be seen in your example image).