You could definitely use turquoise for your back splash. It would also work in your living room as well. Here are some ideas:

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Turquoise would be great in there! But make sure to bring a couple samples of the backsplash tile home with you to take a look at it. I have worked with turquoise and a similar grey-ish color that you seem to have on your walls and that grey can be very tricky as the turquoise likes to turn more green at times! With the grey wood flooring you have, I would hesitate to place black granite in there. Consider looking at taupe or a calcutta quartz to not shock your eye, but allow you to focus a little more on the beautiful design elements that you have in your kitchen including your bay window. As far as the backsplash, try a simple white subway tile (that never goes out of style!) and then having a small inset of turquoise tile in the upper 1/3 part of the backsplash. For you sofa, pull a neutral but make sure you look for a polyester or a polyester/acrylic blend as the fabric will wear much better than some others! Then you can pull fun turquoise accents in throw pillows that have a bold pattern mixed with some solid white pillows as well. Then in your curtains you can pull in a nice patterned drape coordinating with the pillows and dressing up your family room. Hope that helps, and please don't hesitate to reach out should you have any other questions! Best of luck! You're off to a great start!

I would suggest a Carrara marble countertop. A lower cost alternative is a quartz with a Carrara look. Backsplash with contrasting gray grout. Keep it classic. Similarly, I would choose neutral colored large pieces of furniture, and use turquoise accent pieces (throws, pillows, accessories.) I like the "Transitional California" photo that sewsmore posted, showing the two blue ottoman/stools.

Here's a great kitchen with white cabinets, marble countertops and beautiful handmade glazed turquoise terra-cotta tiles in the arabesque shape. One of my favorite kitchens!

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I always like the backsplash to have some connection to the rest of the kitchen I would not do black couter in your space it is too dark I agree a cararra like counter and then a backsplash like this might work

The grey arabesque tile (3rd row, 2nd from top from your last post) would still be neutral but the pattern would add pizzazz. (Assuming you coordinated that with a cararra type of counter). Whatever you decide bring a sample home to see if it works with your paint color. Good luck!

I would avoid doing granite since it's a very busy stone, which can quickly overwhelm your beautiful new kitchen. Try something in grey with maybe a butcher block island to warm up the space. You can add hints of turquoise with a soft glass subway tile in "ocean glass" color (last photo).

(Ceasarstone countertop)

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(Butcher block island with white marble countertop)

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Can you share with us, what color are the walls and what is the floor?