As pointed out by Jason Starr, the answer to your last question is *yes*, so the answer to the question in your title is *no*. Let me give a quick (and straightforward) proof in the case $k= \mathbb{C}$.

First of all, any projective variety over $\mathbb{C}$ has the homotopy type of a finite CW-complex, hence all (co)homology groups of $X$ with coefficients in $\mathbb{Z}$ are finitely generated abelian groups.

Next, let us consider the exponential sequence $$0 \to \mathbb{Z} \to \mathcal{O}_X \to \mathcal{O}_X^* \to 0.$$
Passing to cohomology, and using the assumption $q(X)=0$, we deduce that $\textrm{Pic}\,X=H^1(X, \, \mathcal{O}_X^*)$ injects into $H^2(X, \, \mathbb{Z})$.

Since the latter is a finitely generated abelian group, the same holds for $\textrm{Pic}\,X$.