Sorry but you need to provide a little more info about your scenario.
How are you handling internal routing, between VLANs and forwarding traffic to the Internet?
Reading your question I think that in each PC within each different VLAN, have a (default) gateway which might be the cisco 3560. If so you only be able to interconnect each VLAN. No internet browsing.
To enable each VLAN to connect to the Internet, you need add a default gateway in each one that points to your Internet connected device (Fortigate Firewall ?)
Please let us know if you need further details.