Good thing, if your laptop is working fine on the very same connection as your desktop computer it tell that the problem is on your computer and not on the "network".
The next thing to if to find what block yourself to connect from Google servers. So there is a software or a bad configuration on your computer that prevent the connection to some Google servers.
You could install WinAudit (http://www.parmavex.co.uk/winaudit.html) on your computer and send me the logfile by private message soo I could analyze it in my spare time to try to find the offending software/configuration.
Or a more radical alternative would be to reinstall windows and after install the software you use on by one and test between each installation to find the culprit.