Fwiw, I just mixed faux-marble quartz (Lagoon) with a Carrera backsplash and at least in it's pre-grouted state, I'm really happy. It was hard for me to find a backsplash that worked well with the grey-ish tones of the Lagoon, mostly because my cabinets are painted taupe. Anything that "matched" the cabinets seemed too brownish for the counters. The Carrera blends the two perfectly. I think it also works because I wound up using a very small-scale herringbone pattern, so the patterns in the marble tiles are less pronounced. Fwiw, I don't see the calacutta tiles that the OP posted working with a greyish quartz that's similar to Lagoon, and I'd also be concerned about pattern overload with a subway tile. But having worried about the fake-next-to-real problem, I wanted to at least add my experience that it can be done. One of the prettiest kitchens on here, done by a pro, is GW user @kompy, who used a different marble subway with Lagoon counters, and it looks stunning.

Yes, we need a better picture of the BS. First, I agree with beeps about the 4" BS. My feeling is to do one or the other, but not both. Your issue is something that I faced with my kitchen. My Dove White cabinets were too white for the browns in my Giallo. I first had a white tile installed, but then removed it. Too white for the browns. My current tile is Grazia Rixi in crema. It works as a transition color between the white in the cabinets and the browns in the counter. Your BS picks up on more of the browns and it slso looks a bit busy with your counter. I would find a lighter toned beige or cream tile without the color variation. I would remove the 4" BS and replace with tile.

I think it's best to focus on getting a quartz that is either very solid and/or in a contrasting color to your marble (a distinct shape like herringbone also helps). If they are too close in pattern, the faux finish will stick out, imho. Regarding the backsplash behind the stove -- some people have had problems, especially those that fry a lot. I've been diligent with sealing and wiping up after use. (As aside, we have pure white quartz in one kitchen and gloss black granite in another -- both show crumbs and smears, for different reasons. The grey quartz shows the least).

Thank you to everyone who responded, very much appreciate your iput. The owners of the company agreed to replace the counter at their cost and do it correctly - this was achieved only because we could back up our claims of an incorrect install based on the information provided by you all! Sammy - unfortunately, my sister lost her two dogs and cat in the fire; she was not home. The cause of the fire was the slight reverberations of the built-in microwave caused a screw in the mounting bracket to slowly over time rub the the casing off an electrical wire in the wall; the fire investigator had never seen this before and is now using this in their training. She would trade all of her possessions to have her pets back, a year later it's still devastating for her to have lost them. Again thanks to all and hope everyone has a happy, healthy, safe and prosperous New Year!