In this paper, the Lie group G_NC^α,β,γ, of which the kinematical symmetry group G_NC of noncommutative quantum mechanics (NCQM) is a special case due to fixed nonzero α, β and γ, is three-parameter deformation quantized using the method suggested by Ballesteros and Musso in "Quantum Algebras as Quantizations of Dual Poisson-Lie Groups" [J. Phys. A: Math. Theor., 46 (2013), 195203]. A certain family of QUE algebras, corresponding to G_NC^α,β,γ with two of the deformation parameters approaching

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... ero, is found to be in agreement with the existing results of the literature on quantum Heisenberg group. Finally, we dualize the underlying QUE algebra to obtain an expression for the underlying star-product between smooth functions on G_NC^α,β,γ.