the top is probably just attached with silicone, or metal clips, so you should be able to remove it.

Yes, you are doing to have to remove it, you can't just slip new cabinet beneath an installed countertop.

The countertop is probably glued with silicon. Which means that it is not going to be easy to remove. Silicon really sticks. We generally remove a top by demolishing the center cabinets of a run, supporting the top at both ends, then support the cabinet with a temporary brace while we remove one end cabinet. Then you should be able to lever the top up which will release it from the other end cabinet, but have someone standing buy with a saw to slice through the silicon seal if required.

Good luck with this, it's not easy, and the top will be very heavy. Start with a small top first while you get the hang of it.

OldTimeCarpenter1: Thanks for the information. Probably not a good idea to chance it with there being a 4" backsplash and mirror right on top of it. Would a carpenter be able to add under mount or side mount drawer glides and a soft close element to the drawers?

You probably cannot add gliding hardware to existing drawers, but any cabinet shop can make new drawers and gliding hardware and install them.

Do this the easy way. Imaging the cabinets without doors or drawers -- or, if you are imagination deprived like me -- draw them to scale (still have your old high school drafting kit?). Then decide if you want doors or drawers and what size based on what you are going to store in them.

Treat the existing cabinets like open-faced empty boxes, and start with that.

Quartz has much more strength in tension than granite. It is very unlikely that you will damage it by driving a stiff scraper between the top of the cabinets and the bottom of the tops.

You can have new cabinets installed for as much as someone is going to charge you to repair the old ones.