Washer and dryer in a recessed nook - how do I support a countertop above?
The countertop guys will laugh at you if you want to span quartz over a 5' span.  Basically quartz and granite can barely span a dishwasher width - meaning we get buy with not having plywood over the counters.  And that is given nothing heavy is stored on it on that section.  Given you will have a washer and dryer below it, I would protect it.  Since if a washer or dryer was jerked out of position it could nick or crack the slab.

So I would basically frame out the area.  Cleat on each side - 2x4, then 3 2x4s going the width with one in the middle.  Then a think piece of plywood on top (3/8").  You can then install decorative trim on the front panel and since you used a 2x4, the big box stores will have lots of choices.  Total cost is about $25.  *You can also just face it with a nice wood veneer.  

When the installers come they can just glue it to the plywood - they don't have to overdo this as a couple of dabs will do.  The same principle should be used for the 2x4 framing under.  Meaning if you need to move the quartz up or down or where ever later it should just take a few minutes with a putty knife.  I am sure there is plumbing in that wall and you don't want a shelf you can never move.

Also a couple of notes here:


stone hardly ever looks good floating.  It would look tacky just stuck on top of cleats.  First it isn't thick enough for the floating look and second the underside is hardly ever polished and thinking that would be pretty apparent in its location.
You could also put some drawers below this but this may take some custom cabinetry.