Can I use a sander or grinder to polish quartz countertop edges?
I have only used tools designed specifically for this purpose. For example, Makita makes a stone polishing tool. It looks like an angle grinder, but it has different attachments, variable speed control, and a built in water supply. I am not sure if you can dry sand the quartz, I would imagine it would be extremely dusty. If you're going to wet-sand, you'd have to find a way to protect the surrounding area from water spray.
I think a significant difference between a regular angle grinder and a polisher is the speed. The tool I mentioned earlier can vary speed between 2000 and 4000 rpm, while single speed angle grinders run at a constant, typically higher RPM in the 5000-10000 range. If your angle grinder has variable speed, it might be OK.
And yes, you would want to use stone polishing disks instead of regular sand paper. You would probably wear through the sand paper rather quickly. You can easily find kits of these online for less then $50. They would include all the steps you need (50, 100, 200, 400, 800, 1500, 3000 for example). If you're just doing these edges, you will barely use the pads and can maybe even sell them to someone looking to do a similar project when you're done.