We compute the representation and class counting zeta functions for a family of torsion-free finitely generated nilpotent groups of nilpotency class 2. These groups arise from a generalization of one the families of unipotent groups schemes treated by Stasinski and Voll in [18], [19] and Lins in [10]. The univariate zeta functions are obtained by specializing the respective bivariate zeta functions defined by Lins in [9]. These are also used to deduce a formula for a joint distribution on Weyl groups of type B.