Let $ν_{α,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(α,q)$-Gaussian process (the Gaussian process of type B) over the $(α,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $ν_{α,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation relations satisfied by $(α,q)$-operators are presented.