That's the role the meson (a quark-antiquark pair particle) carries out, but I agree it's confusing. Here's a question from physics stack exchange (without any really clear answers, other than "go look up 'residual strong force'", not very helpful) that spells it out:

> "I just read somewhere that both gluons and mesons transmit the strong force, gluons between quarks inside hadrons, but mesons between nucleons. I thought that the strong force would have one field, and one associated particle, whether inside hadrons, or between nucleons"

How nucleons (neutrons and protons) are bound together is similar to a molecule. If they are close enough, they can 'share' their constituent quarks. You can calculate the interaction by a feynman diagram where the two nucleons exchange one quark in each direction. This is technically the same as one nucleon sending and the other absorbing a quark-antiquark pair, which is why physicists like to say that the nucleon attraction is transmitted by mesons (quark-antiquark pairs). Of course fundamentally the strong force still facilitates the whole interaction, as it's the one preventing nucleons from just falling apart into quarks.