The Chowla conjecture states that if
$t$
is any given positive integer, there are infinitely many prime positive integers
$N$
such that
$\text{Per}\left( \sqrt{N} \right)\,=\,t$
, where
$\text{Per}\left( \sqrt{N} \right)$
is the period length of the continued fraction expansion for
$\sqrt{N}$
. C. Friesen proved that, for any
$k\,\in \,\mathbb{N}$
, there are infinitely many square-free integers
$N$
, where the continued fraction expansion of
$\sqrt{N}$
has a fixed period. In this paper, we describe all polynomials
$Q\,\in \,{{\mathbb{F}}_{q}}\left[ X \right]$
for which the continued fraction expansion of
$\sqrt{Q}$
has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials
$Q$
such that
$\deg \,Q=\,2d$
and
$Per\sqrt{Q}\,=\,t$
.