I will present a construction of the $q$-deformed W-algebra of $\mathfrak{gl}_r$ and its Verma module, that does not use the free field realization or screening charges. The upshot is that our method allows us to directly compute the commutation relations between the Carlsson-Okounkov Ext operator on the moduli space of rank $r$ sheaves on $\mathbb{C}^2$ and the defining currents of the $q$-deformed W-algebra. This implies a geometric representation theory interpretation of the AGT-W relations. Moreover, we will present ideas for generalizing this construction to moduli of sheaves on other surfaces.