  In this paper, we deal with the $f(R,Q)$ gravity whose action depends,
besides of the scalar curvature $R$, on the higher-derivative invariant
$Q=R_{\mu\nu}R^{\mu\nu}$. In order to compare this theory with the usual
General Relativity (GR), we verify the consistency of G\"{o}del-type solutions
within the $f(R,Q)$ gravity and discuss the related causality issues.
Explicitly, we show that in the $f(R,Q)$ gravity there are new G\"{o}del-type
completely causal solutions having no analogue in the general relativity. In
particular, a remarkable G\"{o}del-type solution corresponding to the
conformally flat space and maximally symmetric for physically well-motivated
matter sources, with no necessity of cosmological constant, has been
considered. We demonstrate that, in contrast to GR framework, $f(R,Q)$ gravity
supports new vacuum solutions with the requirement for the cosmological
constant to be non-zero. Finally, causal solutions are obtained for a
particular choice $f(R,Q)=R+\alpha R^2+ \beta Q$.
