Take $(a,b)=(F_{2k},F_{2k+1})$ where $F_0=0$, $F_1=1$ and $F_{n+2}=F_{n+1}+F_{n}$. Then $$(F_{2k}^2+1)(F_{2k+1}^2+1)(F_{2k+2}^2+1)=(F_{2k-1}\cdot F_{2k+1})(F_{2k-1}\cdot F_{2k+3})(F_{2k+1}\cdot F_{2k+3})=(F_{2k-1}F_{2k+1}F_{2k+3})^2$$