The answer is 'no'. If $k = \mathbb{R}$, then $Pic^0(X)(\mathbb{R})$ is a commutative real Lie group of dimension $g$, isomorphic to $(\mathbb{R}/\mathbb{Z})^g$$(\mathbb{R}/\mathbb{Z})^g \times (\mathbb{Z}/2\mathbb{Z})^c$ for some $0 \le c \le g$, so is not finitely generated as an abelian group.