$\DeclareMathOperator\Pic{Pic}$Let $G$ be be a connected reductive group over a field $k$ of characteristic 0. Let $F\subseteq G$ be a connected algebraic subgroup, and set $Y=G/F$.
By Proposition 6.10 in Sansuc's paper [Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002198746&physid=phys48#navi) there is a natural exact sequence of abelian groups
$$
{\sf X}^*(G)\to {\sf X}^*(F)\to \Pic Y\to \Pic G,
$$
where ${\sf X}^*(G)$ denotes the character group of $G$, 
and the map ${\sf X}^*(G)\to {\sf X}^*(F)$ is the restriction homomorphism.
In your case $G=\operatorname{GL}(n)$, and hence $\Pic G=0$. 
We obtain a canonical isomorphism
$$
\Pic Y\cong \operatorname{coker}\left[{\sf X}^*(G)\to {\sf X}^*(F)\right].
$$