Yes, you can export the image out from Surfer. Once you click OK in the Download Online Maps dialog, and the image is downloaded and imported into Surfer as a Base(raster) map, it is like any other image file. There is no longer any link to the server it was downloaded from.
Once it is imported as a base map, you can turn off the axes and export it out to a new georeferenced image file for use in other software if you like.
Updated August 24, 2020