What I'm thinking is this: Let $a = 2n+b$, then the equation becomes $2(2n^2+2bn+b^2)=c^3$. I'm stuck here.
After this, how can I show that $a+bi$ and $a-bi$ are coprime Gaussian integers?
As $(a,b)=1$ both can not be even together
If both are odd $(2A+1)^2+(2B+1)^2\equiv2\pmod8$
$\implies c^3$ is even $\implies c$ is even $\implies c^3\equiv0\pmod8\not\equiv2$
So, $a,b$ must be of opposite parity.
WLOG $a$ is even and $b$ is odd
$\implies a^2$ is even, $b^2$ is odd