Most stone or quartz slabs are sold through a supplier. They will then have to be fabricated to your exact specifications. I would start with looking for a local stone countertop fabricator.

Go to your local Fabricator and look for a remnant. Most fabricators will have lots of remnants that may fit the size you need. Just have your dimensions with you.

Remember that:

1. You will have to support the slab or shelf, and the sink. That will require some kind of bracket which will be visible unless you find some way to hide it inside the wall or within the shelf.
2. The slab and sink will be heavy. You will probably have to install backer board to support them.
3. Vanities hide the wall's water and drain pipes so you can use various elbows, pipes, and other plumbing parts to connect everything up while hiding the mess inside the vanity.
4. The plumbing will be visible so you will probably want to use more expensive metal traps and other plumbing components.

Vanities make it much easier.

A specialty wall hung stone console like that cannot he had by shopping from the cheap remnant like. The logistics and fabrication of such a thing requires the b st fabticator in town, plus s really good plumber, and someone who understsnds it all to design and project manage all the moving parts that it has.