Statues may be built once your Gumball has completed his or her trial in the Fane Expansion. This trial is unlocked once they reach 5 stars and equipped a Certificate. (note, a weapon isn't needed, but highly recommended)
Statue construction time may be reduced by the Architect gumball.
For a complete overview of all Gumballs with a Trial challenge (or simply able to build a statue), refer to the individual gumball pages or see:
See Category:Gumballs with Easter Eggs for a full list of all statue/gumball related easter eggs.