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Using square bridge position, Akbulut-Ozbagci and later Arikan gave algorithms both of which construct an explicit compatible open book decomposition on a closed contact $3$-manifold which results from a contact $(\pm 1)$-surgery on a Legendrian link in the standard contact $3$-sphere. In this article, we introduce the ``generalized square bridge position'' for a Legendrian link in the standard contact $5$-sphere and partially generalize this result to the dimension five via an algorithm which constructs relative open book decompositions on relative contact pairs., Comment: 29 pages, 17 figures