  We derive the quantum analogue of a B\"acklund transformation for the
quantised Ablowitz-Ladik chain, a space discretisation of the nonlinear
Schr\"odinger equation. The quantisation of the Ablowitz-Ladik chain leads to
the $q$-boson model. Using a previous construction of Baxter's Q-operator for
this model by the author, a set of functional relations is obtained which
matches the relations of a one-variable classical B\"acklund transform to all
orders in $\hbar $. We construct also a second Q-operator and show that it is
closely related to the inverse of the first. The multi-B\"acklund transforms
generated from the Q-operator define the fusion matrices of a 2D TQFT and we
derive a linear system for the solution to the quantum B\"acklund relations in
terms of the TQFT fusion coefficients.
