  In this paper, we discuss new results related to the generalized discrete
$q$-Hermite II polynomials $ \tilde h_{n,\alpha}(x;q)$, introduced by Mezlini
et al. in 2014. Our aim is to give a continuous orthogonality relation, a
$q$-integral representation and an evaluation at unity of the Poisson kernel,
for these polynomials. Furthermore, we introduce $q$-Schr\"{o}dinger operators
and give the raising and lowering operator algebra corresponding to these
polynomials. Our results generate a new explicit realization of the quantum
algebra $\mathsf{su}_{q}(1, 1)$, using the generators associated with a
$q$-deformed generalized para-Bose oscillator.
