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 and it is like any other image file. There is no longer any link to the server it was downloaded from.
Once it is imported, 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 November 08, 2017