This message was posted on behalf of Intel Corporation
you can try to disable the "Allow the computer to turn off this device to save power" on 82579LM properties in device manager. also, plug in the LAN to switch port on where ASUS P8Q77-M doesn't experience the issue. There might be configuration on the switch that enables power saving on ports.
I'd also recommend to contact Gigabyte given the ASUS P8Q77-M with the same LAN model didn't encounter the issue. As our LAN controller being integrated to their system manufacturer and further customization might have been done.
Reference site regarding support for Intel® 82579 Gigabit Ethernet Controller