The study of cyclic contraction ordered pairs has garnered significant attention in recent years. We introduce the cyclic ϑ-quasi-contraction ordered pair ((I, ¯I), (J , ¯J )) and investigate the existence and uniqueness conditions for their coupled best proximity points. Furthermore, we present a priori and a posteriori estimates of the best proximity point error of this class of cyclic contraction ordered pairs, particularly highlighting important results for cyclic ϑ-quasi contractions in a uniformly convex Banach space with a modulus of convexity. In Paper [Zlatanov, B. (2016). Error estimates for approximating best proximity points for cyclic contractive maps. Carpathian J. Math.], the author employs geometric progressions for estimations, thus necessitating the condition that the modulus of convexity is of power type, which prevents the generalization of his results to cyclic contractions. The proofs presented in this paper enable error estimation without recourse to geometric progressions. The error calculation in an arbitrarily uniformly convex Banach space is performed solely utilizing its modulus of convexity function, and an error estimation for cyclic ϑ-contraction is provided. Consequently, this work answers the questions posed in the aforementioned paper due to B. Zlatanov.