I'm going to start from your last question. You'll definitely want to use silicone, color-matched if possible, to do the seam between your finished quartz and the first tile of your backsplash. That will help with water resistance and also give you some flex.

You'll probably be fine with silicone underneath to secure the quartz to the brackets, but I wouldn't count on silicone raising the quartz to the appropriate level. You either need to re-position your brackets slightly or shim it to get it to the best height where it will stay. Silicone could flex and strain the seams if your quartz piece isn't securely set at the right height.

As for finishing the two seams, a pro would use something that would be near permanent and very difficult to remove in the event of future problems, also something color matched that would be inconspicuous. They'd also have equipment to pull the sides of the seams together tightly, but since you're working with two pieces on the sides that are previously affixed, you won't be able to do that. There is some type of solution I've seen discussed in these forums that can be ordered color matched for this job, but it would take a while to go back through my saved information and find it. Maybe Joseph Corlett or someone else will see your post and comment. If not, let me know, and I'll try to find it for you. You might want to just do the seams with silicone for now so that if anything else goes wrong, you can re-do the job without damaging all the pieces of quartz trying to take it apart again.

Thank you, Belaria! Very helpful.

Regarding your comment about #1,

A) if the slight difference in level doesn’t bother me aesthetically, is it nevertheless important to make sure the pieces are at exactly the same level?

B) If I did want to try to level the pieces, I’m hesitant about trying to do so by adjusting the brackets, because the house was built 1896 and I’d put money on the wall being slightly angled such that the horizontal piece of the bracket would remain not level no matter where I put it. So witj shims, is there some special type of shim for this type of material? Only shims I’ve used have been the wood ones when framing a house. I can’t picture how I would secure a shim here, either to the metal bracket or to the quartz. Thoughts?

"A) if the slight difference in level doesn’t bother me aesthetically, is it nevertheless important to make sure the pieces are at exactly the same level?"

No.

Set a 4' straightedge across the piece. Shim between the brackets and bottom of the piece until the piece touches the straightedge and hot melt or silicone the shims in place. I'd silicone the seams and back wall and be done.

@Stefan M I've heard of people using playing cards to shim. How big is your gapping? I wasn't thinking of re-drilling the supports necessarily, just wiggling up the brackets if you had any room where the screws are set in.

@Joseph Corlett, LLC, thank you! So what would you recommend I use to shim? Belaria's playing cards idea is interesting . . . curious to know if there's something different you would use.

@Aglitter, I'd say the worst spot of the gap is definitely no more than 1mm, on the right seam toward the front. Either the wall is not square or the adjacent countertop isn't level, because the right seam slopes down ever so slightly. Thanks for the suggestions!

I like small and/or broken pieces of plastic laminate. It's non-compressive, dimensionally stable, and they are small and can be stacked.

Many thanks to you both!