The computer's power supply is likely grounded to the computer chassis on both the AC line input side and on the DC output side.
The driver's power supply chassis is probably also directly connected to the G terminal on the AC line input. So by bolting the power supply chassis to the PC case chassis, you will be sharing that ground connection between the two power supplies. That ground is there for protection against possibly-dangerous wiring faults with the AC line. I do not expect sharing those grounds to cause any issues for you.
The negative terminal of the driver's power supply can be kept isolated from the chassis ground. It goes to the G540's negative power input (and to the limit switches).