The closest thing I found in Diophantus is problem IV(24) which is solving the system $$X_1+X_2=a,\quad X_1X_2=Y^3-Y.$$ Diophantus sets $X_1=x$ and eliminates $X_2$ obtaining $$x(a-x)=Y^3-Y.$$ This is your equation if we set $a=i$ and $y=ix$. This seems to be the first elliptic curve encountered in the book of Diophantus; before that he only considers rational curves and surfaces.
Diophantus choses $a=6$ and obtains a solution $x=26/27,\; Y=17/19$.
(This little research is based on a Russian translation of Diophantus with comprehensive comments by I. G. Bashmakova, published in Moscow in 1974.)