  At the LHC collision energies, multiple parton interactions (MPI) are a key
ingredient for particle production models including hard scale processes like
heavy-quark production (charm and beauty). Quarkonium measurements in
high-multiplicity proton-proton (pp) collisions can shed light on the role of
MPI at such hard momentum scales, as well as on the interplay between hard and
soft particle production mechanisms. In addition, quarkonium production
measurements in minimum bias pp collisions, besides serving as a reference for
heavy-ion collisions collected at the same center-of-mass energy, represent a
benchmark test of various QCD based models. In this contribution, the latest
quarkonium measurements performed by the ALICE collaboration in pp collisions
at several center-of-mass energies are presented. A comprehensive study of the
multiplicity dependence of the quarkonium production at $\sqrt{s} = 13$ TeV,
based on minimum bias and high-multiplicity triggered events, is also
presented. Such measurements include $\psi$(2S) production at forward rapidity
as a function of the charged particle multiplicity density, as well as the
latest multiplicity dependent inclusive J/$\psi$ production measurements at
midrapidity, based on multiplicity estimators covering different pseudorapidity
regions. Similar multiplicity dependent measurements in p--Pb collisions at
center-of-mass energies of $\sqrt{s_{\rm NN}} =$ 5.02 and 8.16 TeV are also
presented. The reported results are compared with available theoretical model
calculations.
