Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D mathcal{N} = 1 supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold S1× ℂ2/ℤp where the action of ℤp is determined by two integer parameters (ν1, ν2). The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of mathfrak{gl} (p). We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of mathfrak{gl} (p). We construct the vertex operator intertwining between these two types of representations. This object is identified with a (ν1, ν2)-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the qq-characters of the quiver gauge theories.