Let $F_k$ be the $k$-th Fibonacci number. Cassini's identity is $F_{k-1}F_{k+1}-F_k^2=(-1)^k$, hence $F_{k-1}^2+F_{k-1}F_k-F_k^2=(-1)^k$. Raising this to the third power and reordering terms yields \begin{equation} 2(F_k^6-F_{k-1}^6)+(F_{k-1}^2-F_{k-1}F_k-F_k^2)^3+(-1)^k=0, \end{equation} which explains infinitely many integer solutions.
See also sequence A337929 at the OEIS.