According to the given information there exist three integers $x,y \in [11,31]$ and $z \in [334,999]$ s.t. $\overline{abc} = x^2$, $\overline{def} = y^2$ and $z^2 = \overline{abcdef} $. Hence $(1) \;\; z^2 = \overline{abcdef} = \overline{abc000} + \overline{def} = 1000 \cdot \overline{abc} + \overline{def} = 1000x^2 + y^2$. By combining equation (1) with the fact that $1000 = 10^2 + 30^2 = 18^2 + 26^2$, we obtain $\overline{abcdef} = (10x)^2 + (30x)^2 + y^2 = (18x)^2 + (26x)^2 + y^2$. An example of such a six digit number is $\overline{abcdef} = 225625$.