Count

16 1by1 +9 2by2+4 3by3 +1 4by4

Formula is 1+4+9+16+..n^2 where n is side length of square(4 in this case)

Counting squares for $n=1,2,3$ we find a pattern for the general case:

$\begin{array} {|l|l|} \hline n & \mathsf{Count }& \mathsf{Total }\\ \hline 1 &1 &1\\ \hline 2 & 1+2.2 & 5\\ \hline 3 & 1+2.2+3.3 & 14 \\ \hline ... & ... & ... \\ \hline n & 1^2+2^2+3^2+...+n^2 & \sum_{k=1}^{n}k^2 \\ \hline \end{array}$

We have a well known summation (the closed form could be derived by finite calculus): $\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}{6}$

Then for $n=4$ :

$\frac{4(4+1)(2.4+1)}{6}=\boxed{30}$

In a standard n*n grid, number of squares is given by $\Sigma n^2$ = $\frac {n(n+1)(2n+1)}{6}$

Note here n = 4.