  Using the homogeneous G\"{o}del spacetimes we find some new solutions for the
field equations of bosonic string effective action up to first order in
$\alpha'$ including both dilaton and axion fields. We then discuss in detail
the (non-)Abelian T-dualization of G\"{o}del string cosmologies via the
Poisson-Lie (PL) T-duality approach. In studying Abelian T-duality of the
models we get seven dual models in such a way that they are constructed by
one-, two- and three-dimensional Abelian Lie groups acting freely on the target
space manifold. The results of our study show that the Abelian T-dual models
are, under some of the special conditions, self-dual; moreover, by applying the
usual rules of Abelian T-duality without further corrections, we are still able
to obtain two-loop solutions. We also study the Abelian T-duality of G\"{o}del
string cosmologies up to $\alpha'$-corrections by using the T-duality rules at
two-loop order derived by Kaloper and Meissner. Afterwards, non-Abelian duals
of the G\"{o}del spacetimes are constructed by two- and three-dimensional
non-Abelian Lie groups such as $A_2$, $A_2 \oplus A_1$ and $SL(2, \mathbb{R})$.
In this way, the PL self-duality of $AdS_3 \times \mathbb{R}$ space is
discussed.
