It is not necessary that the Secretary General should be a citizen of one of the great powers/ the big five.
Here is the list of Secretary Generals with their countries:
Mr. Trygve Lie (Norway)
Mr. Dag Hammarskjold (Sweden)
Mr. U. Thant (Mayanmar)
Mr. Kurt Waldheim (Austria)
Mr. Javier Perez de Cuellar (Peru)
Mr. Boutros Boutros Ghali (Egypt)
Mr. Kofi Annan (Ghana)
Mr. Ban Ki Moon (South Korea)
Mr. Antonio Guterres (Portugal)