Cut the chicken into 1-inch chunks and place in a zip-top bag with 1 tbsp olive oil and 2 tbsp Cajun seasoning; mix to coat and let marinate while you prep the vegetables.
Cut the bell peppers into bite-sized pieces and press the garlic; set aside the chopped green onion for garnish.
Heat 1 tbsp olive oil in a large skillet over high heat. Add marinated chicken in a single layer (work in batches if needed) along with the peppers.
Cook the chicken without stirring for 3–4 minutes, flip pieces with tongs and cook 2 more minutes, then add the pressed garlic and cook 1 additional minute until chicken reaches 165°F.
Remove the cooked chicken and peppers from the skillet and set aside. In the same skillet over medium heat, melt the butter and add 1 tbsp Cajun seasoning.
Add the heavy cream, cream cheese, and grated Parmesan to the skillet and whisk until the cheeses melt and the sauce comes to a brief boil, about 1 minute; then remove from heat.
In a separate large skillet, heat 1 tbsp olive oil over high heat. Add the spiralized zucchini and cook 2–3 minutes without stirring, then stir and cook another 2 minutes until softened but not mushy.
Toss the zoodles, cooked chicken, and peppers with the Alfredo sauce until evenly coated.
Serve immediately and sprinkle chopped green onion over the top.