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To a 2-dimensional finite complex $X_\\Gamma$ arising as the quotient of the Bruhat-Tits building $X$ associated to $\\Sp_4(F)$ by a discrete torsion-free cocompact subgroup $\\Gamma$ of $\\PGSp_4(F)$, associate the zeta function $Z(X_{\\Gamma}, u)$ which counts geodesic tailless cycles contained in the 1-skeleton of $X_{\\Gamma}$. Using a representation-theoretic approach, we obtain two closed form expressions for $Z(X_{\\Gamma}, u)$ as a rational function in $u$. 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