Export a Texture that has been edited via SketchUp's Native Editor

I quite like the way the brick texture looks after I have edited the colors.

How do I make it a new texture?

Make a square face and apply the texture you like to it

If you right click the face and choose “make unique texture” - this will create a new texture in the shape of your face with the texture adjustments applied.
This does however compress the texture a 2nd time, so you can suffer some quality losses.