The measurement of ϒ(1S), ϒ(2S), and ϒ(3S) yields as a function of the charged-particle multiplicity density, dNch/dη, using the ALICE experiment at the LHC, is reported in pp collisions at s= 13 TeV. The ϒ meson yields are measured at forward rapidity (2.5ch/dη, within the uncertainties. The self-normalized yield ratios of excited-to-ground ϒ states are compatible with unity within uncertainties. Similarly, the measured double ratio of the self-normalized ϒ(1S) to the self-normalized J/ψ yields, both measured at forward rapidity, is compatible with unity for self-normalized charged-particle multiplicities beyond one. The measurements are compared with theoretical predictions incorporating initial or final state effects.