The quantum actions of the (4, 4) supersymmetric nonlinear sigma model and
its dual in the Abelian case are constructed by using the background superf
ield 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 s
igma model ia the sense of a Poisson-Lie T-duality.