This paper investigates a natural linear family of cyclic Latin squares parametrized by the Fibonacci sequence over Z_N. We provide explicit greatest common divisor (gcd) criteria for Latinity, the existence of affine transversals, and mutual orthogonality. By leveraging classical Fibonacci identities (d'Ocagne and Cassini) and the Chinese Remainder Theorem, we show that consecutive arrays in this family naturally form orthogonal pairs. Furthermore, we derive an exact multiplicative counting formula for the total number of affine transversals, demonstrating explicitly that no such transversals exist for even moduli.