Licensing: Please see the Space Engine site for information on licensing. I believe that by buying the basic Space Engine you do NOT acquire a license to use exported images.
However, you also can only export very low rez skybox images that look pretty crappy, so for testing and development I am just using them for now.
To obtain a license to use these images in your work and the ability to export high resolution images look for and buy the "SpaceEngine PRO" license.

Export Skybox from Space Engine

• Find a spot in the Universe where you like the view.
• Tick things off/on that you don't want to see like labels or galaxies or whatever.
• Escape to the main menu and select:
  • Tools, then
  • Export skybox
• Note that you are still "in the engine" now so you can still move the view around and tick features on and off.
• Turn OFF "Identify faces" as you don't want that text in the image.
• Export
  Space engine will just put the files in this folder, overwriting the last save:
  C:\Program Files (x86)\Steam\steamapps\common\SpaceEngine\export

Convert the Cubemap to a "Panorama"

As per the Godot documentation it uses a "equirectangular sky map" instead of a cube map and you can use this tool to convert it:
https://docs.godotengine.org/en/stable/classes/class_panoramaskymaterial.html#class-panoramaskymaterial

Upload each segment form the folder. The names of the files should make it clear which one goes where.
Set the rotation to 0 for ALL segments.
use the "Update" button to update the image.
When you are done use "Download(save) result" to get the file for Godot.

Copy this saved file into your Godot project somewhere.

Set up a Skybox in Godot

• Add a "WorldEnvironment" to your scene

• Add a "New Environment" to it
  

• Click on the Environment and under Background -> Mode select "Sky"
  

  • Open up the "Sky" drop down now and click on <empty> and add a new Sky
    

• Click on "Sky" to make it open up and click on <empty> underneath next to "Sky Material" and select "New PanoramaSkyMaterial"
  

• Click on "PanoramaSkyMaterial" to make it open up so that you see <empty> next to "Panorama"

• Drag the file panorama image file you saved from the web site earlier onto this box.
  

Now it should look something like this and you will see the exported background from Space Engine in your editor.