I got the "No Internet Conection" message when tried to update my C7 to Belle. I've already re installed Nokia Suite to version 3.3.86, disabled firewall and still nothing. I don't use any proxy and already checked the certificates issue and all is fine.
Can you browse the internet using Internet Explorer?
Have you tried uninstalling Nokia Suite from the PC which doesn't work and reinstalling it?
Also, it may be worth to reset your router and change your password, then reconnect. Do this before reinstalling the application, to make sure it is not a network issue.
If the problem persists check your firewall settings.
You can try also turns out firewall and try to uninstall the antivirus and then restart computer. After computer came on again, I hope that the updater for nokia belle will work.