Cook the quinoa according to the package directions. Keep covered to keep warm.
Meanwhile, heat a large skillet over medium heat. Season the chicken with ¼ teaspoon salt and ¼ teaspoon pepper. Cook, skin-side down, without moving, until the skin is browned and crisp, about 15 minutes. Reduce heat to low, flip the thighs, and continue cooking until an internal thermometer inserted into the thickest part of the thigh registers 165° F, another 15 minutes. Remove from the pan and keep warm.
Drain all but 2 tablespoons of fat from the pan and return to medium heat. Add the green beans and ¼ cup water and bring to a simmer, scraping free any browned bits. Cover and cook until crisp-tender, about 8 minutes. Uncover and add the spinach. Cook until wilted, 1 to 2 minutes. Add the butter and stir to melt. Toss in the giardiniera and brine and season with salt and pepper.
Serve the chicken, vegetables, and pan sauce over the quinoa.