  We study the Virasoro constraints for moduli spaces of representations of
quiver with relations by Joyce's vertex algebras. Using the framed Virasoro
constraints, we construct a representation of half of the Virasoro algebra on
the cohomology of moduli stacks of quiver representations under smoothness
assumption. By exploiting the non-commutative nature of the Virasoro operators,
we apply our theory for quivers to del Pezzo surfaces using exceptional
collections. In particular, the Virasoro constraints and representations are
proven for moduli of sheaves on $\mathbb{P}^2$, $\mathbb{P}^1\times
\mathbb{P}^1$ and $\text{Bl}_{\mathsf{pt}}(\mathbb{P}^2)$. Lastly, we unravel
the Virasoro constraints for Grassmannians in terms of symmetric polynomials
and Hecke operators.
