  In Lema\^{\i}tre -- Tolman (L--T) models, light rays emitted
\textit{radially} at the Big Bang (BB) at such radial coordinates $r$ where the
bang-time function $t_B(r)$ has $\dril {t_B} r \neq 0$ reach every observer
with \textit{infinite blueshift}, $z = -1$. Consequently, there exist rays,
emitted soon after the BB, that will reach later observers with finite
blueshift ($-1 < z < 0$). But in spacetimes without symmetry there are no
radial directions. The question thus arises whether blueshifts can exist at all
in the Szekeres models that contain L--T as a limit, but in general have no
symmetry. The aim of the present paper is to show that strong blueshifts can be
generated in quasi-spherical Szekeres (QSS) models. It is shown that in an
axially symmetric QSS model, infinite blueshift can appear only on axial rays,
which intersect every space orthogonal to the dust flow on the symmetry axis.
In an exemplary QSS model it is numerically shown that if such a ray is emitted
from the Big Bang where $\dril {t_B} r \neq 0$, then indeed observers see it
with $z \approx -1$. Rays emitted shortly after the BB and running close to the
symmetry axis will reach the observers with a strong blueshift, too. Then, in a
toy QSS model that has no symmetry, it was shown by numerical calculations that
two null lines exist such that rays in their vicinity have redshift profiles
similar to those in a vicinity of the axial rays in the axially symmetric case.
This indicates that rays generating infinite blueshifts exist in general QSS
spacetimes and are concentrated around two directions.
