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We construct the twisted Fock module of quantum toroidal $\mathfrak{gl}_1$algebra with a slope $n'/n$ using vertex operators of quantum affine$\mathfrak{gl}_n$. The proof is based on the $q$-wedge construction of anintegrable level-one $U_q(\widehat{\mathfrak{gl}}_n)$-module and therepresentation theory of double affine Hecke algebra. The results areconsistent with Gorsky-Negu\c{t} conjecture (Kononov-Smirnov theorem) on stableenvelopes for Hilbert schemes of points in the plane and can be viewed as amanifestation of $(\mathfrak{gl}_1,\mathfrak{gl}_n)$-duality.

Twisted Fock module of toroidal algebra via DAHA and vertex operators