  The Georgi-Machacek (GM) model is a distinctive TeV-scale extension of the
Standard Model (SM) due to the introduction of two (one real and one complex)
scalar triplets to the Higgs sector. It predicts the existence of doubly
charged Higgs bosons $H_5^{\pm\pm}$ and the vacuum expectation value of Higgs
triplets $v_\Delta$ can reach a few tens of GeV. In this paper, we perform a
parameter scan of the GM model within the H5plane benchmark scenario and
investigate in detail the single production of a doubly charged Higgs boson in
association with a SM-like Higgs boson via vector-boson fusion at the $14~ {\rm
TeV}$ LHC and $70~ {\rm TeV}$ Super Proton-Proton Collider. Both integrated
cross section and differential distributions with respect to some kinematic
variables for $pp \rightarrow W^{\pm}W^{\pm} \rightarrow H_5^{\pm\pm} h^0 + 2\,
{\rm jets}$ are provided up to the QCD next-to-leading order. In the
signal-background analysis we employ the {\sc madspin} method to take into
account the spin correlation and finite width effects of the intermediate Higgs
and $W$ bosons and present some kinematic distributions of final leptons. The
numerical results show that the SM background can be suppressed remarkably and
the $H_5^{\pm\pm} h^0$ vector-boson fusion signal can be directly detected at
future high-luminosity, high-energy hadron colliders by imposing a proper cut
on the transverse mass $M_{T, \ell_1 \ell_2}$.
