  We study {\em right-invariant (resp., left-invariant) Poisson quasi-Nijenhuis
structures} on a Lie group $G$ and introduce their infinitesimal counterpart,
the so-called {\em r-qn structures} on the corresponding Lie algebra $\mathfrak
g$. We investigate the procedure of the classification of such structures on
the Lie algebras and then for clarity of our results we classify, up to a
natural equivalence, all $r$-$qn$ structures on two types of four-dimensional
real Lie algebras. We mention some remarks on the relation between $r$-$qn$
structures and the generalized complex structures on the Lie algebras
$\mathfrak g$ and also the solutions of modified Yang-Baxter equation on the
double of Lie bialgebra $\mathfrak g\oplus\mathfrak g^*$. The results are
applied to some relevant examples.
