If we assume that in a system heat is only transferred by conduction (between two different solid of different materials), what role do phonons play in this phenomenon?
It depends on what type of system you're dealing with. Are you talking about two macroscopic cubes of different materials that you squeeze together with your hands? Or are you talking about some sort of heterostructure where the different materials are bonded together at the atomic level?
In the first case, it's tricky to think about phonons transferring heat because the details of the surface are going to be so important. I'm guessing it would be some version of the diffuse mismatch model (below), but it only applies where the materials are in very close contact, which depends on how smooth the surfaces are. If the cubes have oxides or other gunk on them, then you have even more materials in play (e.g. the two materials and their two oxides).
In the second case, phonons are very important for heat transfer, but modeling them is kind of tricky. The two simplest models are the acoustic mismatch model (AMM) and diffuse mismatch model (DMM).
The AMM basically treats phonons at the interface the same way you treat light transmitting or reflecting when moving from one material to another. E.g. there's a version of Snell's law for phonons, the transmission depends on the angle and difference in group velocities in the two materials, etc. The AMM is most appropriate for a very smooth interface.
The DMM is more appropriate for a rough or otherwise disordered interface. The DMM basically says that any phonon that hits the interface gets scattered in a random direction by the disorder.
There are much mode advanced models too, especially molecular dynamics (MD) simulations, where the motion of each atom is tracked individually. However, MD simulations require a lot of computer power, even for really tiny systems.
The concept of phonons can be used to describe the thermal conductivity and heat capacity of a material. These are important to determine how much “thermal mass” a material has, and how heat will diffuse over time. But conduction between two materials is primarily determined by the interface. Phonons are not not really the best model for this interfacial physics, which can be complicated. But if you can assume perfect thermal conductance across an interface, then the overall conductivity would depend on just the bulk properties, which phonons determine.
It depends on the type of materials. If you take two metals at "low" temperatures, phonons play only a minor role in the thermal conductivity. In metals, the electronic contribution to the thermal conductivity can be dominant at low temperatures. At higher temperatures, phonons start to have a major role.
For insulators, the thermal conductivity is mainly caused by phonons, so their role to transfer heat between the two materials would be important.
For non doped semiconductors, phonons play a major role in the thermal conductivity while the more they are doped, the more the electronic contribution rises.