Imbed rods in the wall studs and drill the back of the estone. Better stay very level.

I don’t think that is quartz in either of those photos. And I cannot emphasize enough how sticky and gross everything placed on that shelf will get. Who needs to add such a cleaning chore to their life?

Quartz would never survive in this extremely hot place! The resin cannot handle the flame right below it. That must be real stone.

Also, reaching over gas flames for stuff could be a safety issue.

I wouldn’t. For the reasons mentioned above and the potential hazard of something falling off the shelf into the flames or pot or pan.

I wouldn't install the shelf or even attempt it. And I have a downdraft, not hood - also an electric stove not gas. Clearly not a perfect kitchen, but I do well to get the grease/steam spots off the subway tile behind before they've turned in little yellow greasy petrified dots that must be loosened one at a time. I'd never keep up a shelf. I don't know what I'd use there that I could expose to any additional heat or steam from boiling pasta anyway.

Would not put a shelf there.

What @lucky998877 said.

Also note that those ranges are installed with "island" trim, not factory backsplashes recommended for this install unless the backsplash is a non combustible assembly.