The number $2010$ is a multiple of $3$ but not of $9$. The sum of two squares can be a multiple of $3$ only if the two squares involved are both multiples of $3$ – which means they must be multiples of $9$ as well. So $2010$ is not the sum of two squares.

One can write $2010$ in various ways as the sum of three squares. For completeness, all possible such representations are listed below.
$$\begin{aligned} 2010 &= 1^2 + 28^2 + 35^2 \\ &= 4^2 + 25^2 + 37^2 \\ &= 5^2 + 7^2 + 44^2 \\ &= 5^2 + 31^2 + 32^2 \\ &= 7^2 + 19^2 + 40^2 \\ &= 11^2 + 17^2 + 40^2 \\ &= 16^2 + 23^2 + 35^2 \\ &= 19^2 + 25^2 + 32^2 \end{aligned}$$