  Given a non-negative integer $q$, we study two different notions of the
$q$-capability of Lie algebras via the non-abelian $q$-exterior product of Lie
algebras. The first is related to the $q$-crossed modules and inner
$q$-derivations, and the second is the Lie algebra version of the
$q$-capability of groups proposed by Ellis in 1995.
