Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Melt butter and stir in farro. Cook and stir for about 2 minutes. Pour in chicken stock and stir to combine. Change to Manual setting. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Add cream, mozzarella cheese, Parmesan cheese, and Greek seasoning. Stir until melted.