We are consistent with the notation in the book of Hartshorne.
Let $X=G(\mathbb P^k, \mathbb P^n)$ be the Grassmannian parametrizing $\mathbb P^k$ contained in $\mathbb P^n$. We have the so-called universal sequence of vector bundles over $X$: $$ 0 \to S^\vee \to V \otimes \mathcal O \to Q \to 0 $$ where $V$ is the vector space such that $\mathbb P^n = \mathbb P V$ in the Grothendieck' sense. From this fact it follows that $Q$ is a globally generated vector bundle over $X$. Instead, its dual $Q^\vee$ is not globally generated.
By the classic theory, there exists $m_0 \ge 0$ such that, for all $m \ge m_0$, $Q^\vee(m)$ is globally generated. Is this value $m$ known? Also, when $X$ is an isotropic Grassmannian (orthogonal for example), does the same $m$ work again?
$m = 1$. In fact, $$Q^\vee(1) \cong \wedge^{\mathrm{rank}(Q)-1}Q$$ and a wedge power of a globally generated vector bundle is globally generated. For isotropic Grassmannian the same is true by restriction.