In short: Yes nucleons can exchange quarks via mesons, but because of confinement the baryons will always have three valence quarks.

But all, e.g., up quarks are indistinguishable. So it makes no sense to say that a particular quarks left the proton and travelled to another (as part of a pion for example). At the same time a new up-quark must have been created and remained in the original proton.

And then there is the issue of energy scale. The quarks are still bound inside the nucleons, meaning the energy required to get one out is larger than the random energy fluctuations in the system (= temperature). That is why you only exchange virtual mesons rather than real particles. Mathematically this makes it convenient to describe the nucleons as their own things that than interact with one another, rather than looking at all the quarks and gluons and whatnot separately.

You can create states of matter where the quarks are no longer bound, i.e. it makes sense to model things on the quark level. It is called "quark gluon plasma", and it exists at much higher energies/temperatures. E.g. it is created briefly in particle colliders, when you smash nuclei into each other.