Yes, in principle only certain quasi-momentum are allowed. However, the spacing between allowed quasi-momenta will scale inversely with the number of atoms in the solid. Since most solids contain a lot of atoms, the spacing will be very tiny and the spectrum of quasi-momenta is treated as continuous.
The main difference between phonons and free particles is that momentum must be conserved, but quasi-momentum only has to be conserved up to a reciprocal lattice vector. This leads to umklapp scattering, which can lead to counter-intuitive results. For example two phonons can interact and scatter in the "wrong" direction (as shown in the link).