  In [1], a generalized type of Darboux transformations defined in terms of a
twisted derivation was constructed in a unified form. Such twisted derivations
include regular derivations, difference operators, superderivatives and
$q$-difference operators as special cases. The formulae for the iteration of
Darboux transformations are expressed in terms of quasideterminants. This
approach not only enables one to recover the known Darboux transformations and
quasideterminant solutions to the noncommutative KP equation, the non-Abelian
two-dimensional Toda lattice equation, the non-Abelian Hirota-Miwa equation and
the super KdV equation, but also inspires us to investigate quasideterminant
solutions to $q$-difference soliton equations. In this paper, we first derive
the bilinear B\"acklund transformations for the known bilinear $q$-difference
two-dimensional Toda lattice equation ($q$-$2$DTL), then derive a Lax pair
whose compatibility gives a new nonlinear $q$-$2$DTL equation and finally
obtain its quasideterminant solutions by iteration of its Darboux
transformations.
