This paper introduces and investigates a novel class of skew-regular Quaternary Hadamard matrices. For every odd prime power $p$, we establish the existence of these matrices for all orders $1+p^2$, each characterized by a constant row sum of $1-pi$. Motivated by the growing importance of large-excess matrices in the maximum determinant problem and quantum nonlocality, we demonstrate that this class provides a robust framework for generating Hadamard matrices with exceptionally large excess.