Combine flour, salt, black pepper, and cayenne pepper in a resealable plastic bag and shake to combine. Add calamari and make sure all pieces are coated.
Heat oil in a frying pan over medium to medium-high heat to 395 degrees F (200 degrees C).
While oil heats, combine tomatoes, onion, and celery in a bowl. Set tomato relish aside.
Shake excess flour from calamari and drop calamari into the hot oil in batches. Fry until golden brown, about 1 minute per batch. Drain on paper towels. Transfer cooked calamari to a serving plate.
Top with crumbled blue cheese, blue cheese salad dressing, hot sauce, and a sprinkling of tomato relish.