<p>This paper investigates linear neutral quaternion differential equations with a single constant delay, focusing primarily on right-sided coefficient systems, whose structure differs essentially from the left-sided formulation due to the noncommutativity of quaternion multiplication, while also briefly addressing the left-sided case. Since quaternion multiplication is noncommutative, classical fundamental-matrix and Laplace-transform-based methods are not directly applicable. We introduce a family of <i>recursive neutral quaternion matrix equations</i> and define a corresponding <i>neutral quaternion matrix function</i> that serves as a fundamental solution for neutral quaternion delay systems. Using this function and a variation-of-parameters approach, we derive explicit representation formulas for homogeneous and nonhomogeneous neutral LQDEs, incorporating the initial history and forcing terms via convolution-type integrals. Furthermore, we establish the completeness of the quaternion-valued continuous function space under the supremum norm and prove existence and uniqueness for a nonlinear neutral quaternion system under a Lipschitz condition using the Banach fixed-point theorem. Finally, Ulam–Hyers stability is obtained with an explicit stability constant.</p>