Bring a large pot of salted water to a boil. Add the fettuccine pasta and cook according to package instructions until al dente, about 10-12 minutes. Drain and set aside.
While the pasta cooks, heat olive oil in a large skillet over medium heat.
Season the salmon fillets evenly with salt and black pepper on both sides.
Place the salmon fillets in the skillet and cook for 4-5 minutes on each side, until cooked through and easily flaked with a fork. Remove the salmon from the skillet and set aside.
In the same skillet, reduce heat to medium-low and add the butter. Once melted, add the minced garlic and sauté for 1 minute until fragrant.
Pour in the heavy cream and stir to combine. Let the cream simmer gently for 3-4 minutes until slightly thickened.
Add the grated Parmesan cheese and stir until melted and smooth.
Add the fresh spinach leaves and cook until wilted, about 2 minutes.
Flake the cooked salmon into bite-sized pieces and gently stir into the cream sauce.
Add the cooked pasta to the skillet and toss to coat evenly with the sauce.
Stir in the fresh lemon juice and red pepper flakes, adjusting seasoning with additional salt and pepper if needed.
Serve the creamy salmon pasta immediately, garnished with extra Parmesan if desired.