Hi,
The computer should have 2 NICs. NIC1 is connected to the port in the switch which is a member of the first VLAN while NIC2 is connected to the port in the switch which is a member of the second VLAN.
Depending on the OS, only one of the NIC is routable by default. If you need both NIC to be routable, in Windows you need to run the Routing Service. However, only one can be configured for default route, the other can be configured for more specific route. You maybe able to configure both for default route but think about what that configuration will do.
This setup is common for server which use the first NIC with default route for application (i.e. Web, FTP, etc..) and the second NIC (GE perhaps) for Network Backup.
Old network backup design use the second NIC non-routable and all server including the backup server is in this second NIC broadcast domain. However, this design has a flaw, if a hacker successfuly gain access to any of the server, it can successfully gain access to all the servers in the network backup domain. The new design has a firewall between servers in the network backup domain separating them by segment. With this new design, if routing is not enable in the NIC use for network backup, NAT is use in the segment of the backup server, else you can turn ON routing in the second NIC.
Regards,
Dandy