  We investigate the quasi-Hopf twist of the quantum toroidal algebra of
$\mathfrak{gl}_1$ as an elliptic deformation. Under the quasi-Hopf twist the
underlying algebra remains the same, but the coproduct is deformed, where the
twist parameter $p$ is identified as the elliptic modulus. Computing the
quasi-Hopf twist of the $R$ matrix, we uncover the relation to the elliptic
lift of the Nekrasov factor for instanton counting of the quiver gauge theories
on $\mathbb{R}^4 \times T^2$. The same $R$ matrix also appears in the
commutation relation of the intertwiners, which implies an elliptic quantum KZ
equation for the trace of intertwiners. We also show that it allows a solution
which is factorized into the elliptic Nekrasov factors and the triple elliptic
gamma function.
