  The authors extend to the $q-$tensor square $G \otimes^q G$ of a group $G$,
$q$ a non-negative integer, some structural results due to R. D. Blyth, F.
Fumagalli and M. Morigi concerning the non-abelian tensor square $G \otimes G$
($q = 0$). The results are applied to the computation of $G \otimes^q G$ for
finitely generated nilpotent groups $G$, specially for free nilpotent groups of
finite rank. They also generalize to all $q \geq 0$ results of M. Bacon
regarding an upper bound to the minimal number of generators of the non-abelian
tensor square $G \otimes G$ when $G$ is a $n-$generator nilpotent group of
class 2. The paper ends with the computation of the $q-$tensor squares of the
free $n-$generator nilpotent group of class 2, $n \geq 2$, for all $q \geq 0.$
This shows that the above mentioned upper bound is also achieved for these
groups when $q > 1.$
