If GOG Galaxy is not able to connect while your internet works just fine, try the following steps:
1) Open the Start Menu and type CMD in search bar.
2) Launch the Command Prompt by right-clicking and selecting "Run as Administrator".
netsh winsock reset
in the window that appears and hit Enter to confirm.
4) Wait until the Winsock reset is done, you should see "Winsock reset completed successfully" as confirmation.
5) Close the Command Prompt and restart your PC to finish the process.
In case the issue persists, please contact support here