To each hyperbolic Landau level of the Poincar\'e disc is attached a
generalized negative binomial distribution. In this paper, we compute the
moment generating function of this distribution and supply its decomposition as
a perturbation of the negative binomial distribution by a finitely-supported
measure. Using the Mandel parameter, we also discuss the nonclassical nature of
the associated coherent states. Next, we determine the L\'evy-Kintchine
decomposition its characteristic function when the latter does not vanish and
deduce that it is quasi-infinitely divisible except for the lowest hyperbolic
Landau level corresponding to the negative binomial distribution. By
considering the total variation of the obtained quasi-L\'evy measure, we
introduce a new infinitely-divisible distribution for which we derive the
characteristic function