The quantum actions of the (4,4) supersymmetric nonlinear sigma
model and its dual in the Abelian case are constructed by using the
background superfield method. The propagators of the quantum superfield
and its dual and the gauge-fixing actions of the original and dual (4,4)
supersymmetric sigma models are determined. On the other hand, the BRST
transformations are used to obtain the quantum dual action of the (4,4)
supersymmetric nonlinear sigma model in the sense of a Poisson-Lie
T-duality.