A search for nonresonant Higgs boson ($H$) pair production via gluon and vector boson ($V$) fusion is performed in the four-bottom-quark final state, using proton-proton collision data at 13 TeV corresponding to $138\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected by the CMS experiment at the LHC. The analysis targets Lorentz-boosted $H$ pairs identified using a graph neural network. It constrains the strengths relative to the standard model of the $H$ self-coupling and the quartic $VVHH$ couplings, ${\ensuremath{\kappa}}_{2V}$, excluding ${\ensuremath{\kappa}}_{2V}=0$ for the first time, with a significance of 6.3 standard deviations when other $H$ couplings are fixed to their standard model values.