Does there exists a process of boson-fermion interaction through which a virtual boson that carries force on fermionic matter turns into two (or more) fermions (with the correct conservations)? If not, through which other process can a boson turn into a fermion and vice versa?
A real boson can certainly turn into fermions. This is exactly what happens in pair production when a photon turns into an electron and positron.
But you need to be very cautious when talking about virtual particles because virtual particles don't exist. The Feynman diagrams that we draw showing virtual particles are just a graphical representation of an integral called a propagator and do not show anything that actually happens. We can, and do, draw Feynman diagrams where a gauge boson turns into two fermions, but this is not showing a process that actually happens.