Is it possible to calculate the number of neutrinos that would be produced from the conversion of 1kg of mass to energy?
Mass does not turn to energy magically. It needs a good definition of the problem, for example in fusion or fission, as you ask, or in the cosmological models primordial creation.
Would the calculation be any different if the conversion took place from a fusion reaction compared to a matter-antimatter annihilation?
Of course all three models will need different calculations. Neutrinos from fission and fusion depend on the interactions and the strong nuclear force binding nuclei, and will be different for either case, and only a small part of the mass can be converted to energy. Primordial neutrinos will use the standard model of particle physics plus the nuclear models. Many more particles will be sharing the energy than just neutrinos. Calculations have been done and can be done , depending on the particular boundary conditions.