Nope. Don't do it. Find tile to coordinate with your quartz. Something fairly simple that won't compete with the veining OR do the full backsplash in the same quartz.

P.S. Not in love with your flooring choice. Too busy and not the correct shade of gray. I'd opt for a porcelain tile that looks like charcoal slate, assuming the veins in the quartz are gray. (Hard to tell from these photos.)

Daltile makes a great slate-look porcelain floor tile and it's quite inexpensive.

Thanks for the advice. I have to go with a laminate because I am not removing the original 1950s linoleum (asbestos concerns)

That's fine, but please find a solid option instead of a pattern. Too much going on.