Okay, do you know how to add an exception to the windows firewall? Assuming you don't.
- Right click the start icon, and select the Control Panel option.
- Go to "System and Security".
- Under "Windows Firewall, select "Add an app through Windows Firewall".
- In that list, see if you can find GOG Galaxy Client.
- If you can, tick all the boxes beside it.
- If you can't, Click on "Add Another app", choose "Browse", and navigate to the GalaxyClient executable, and give it the necessary permissions.
I'm assuming this is going to fix your problem.