I'm on a tight budget so this is one of the few option. So, is the RX 580 compatible with the Acer Veriton M6610? If not, is there any other GPU compatible?
If this is the product in question
Model Name: Veriton M6610G
Part Number: PS.VCC03.009
then NO. Do not buy it. Not only is the processor ancient and vulnerable to every Intel processor vulnerability discovered and severely impacted by mitigations, the PSU is far insufficient.
The Q67 is Sandy Bridge which has the Cougar Point problems in addition to the Meltdown and Spectre problem etc.
In short I suggest building a new machine from scratch, starting with a chassis and adequate PSU etc