Can anybody tell if it is possible for an experienced hire to get into one of the consulting firms in Hungary without the knowledge of Hungarian? I am planning to move there and would love to try consulting. Languages: Russian (native), English, Serbian + basic knowledge of Spanish.

There aren't many local consulting firms in Hungary and those do require Hungarian because they focus on the local market. But there are international firms present as well which might accept the lack of Hungarian. I'd try them first.