The classical Bargmann representation is given by operators acting on the space of holomorphic functions with the scalar product $⟨zⁿ|z^k{⟩}_q={\delta }_{n,k}{\left[n\right]}_q!=F\left(zⁿz{̅}^k\right)$. We consider the problem of representing the functional F as a measure for q > 1. We prove the existence of such a measure and investigate some of its properties like uniqueness and radiality. The above problem is closely related to the indeterminate Stieltjes moment problem.