  We discuss the quantization of the $\widehat{\mathfrak{sl}}_2$ coset vertex
operator algebra $\mathcal{W}D(2,1;\alpha)$ using the bosonization technique.
We show that after quantization there exist three families of commuting
integrals of motion coming from three copies of the quantum toroidal algebra
associated to ${\mathfrak{gl}}_2$.
