I count 16. Since 16 is a square (the square of 4), maybe you could say there are 17.
Systematic attack: examine each node to see of how many squares that node is the upper left corner. Add the counts.
Then, as a check, do the same, counting for each node the number of squares for which it is the upper
right corner. Add the counts, and the total should be the same.
Here's an annotated copy. Next to each node I have placed the number of squares of which that node is the upper left corner.
There are nine squares with sides of length 1.
There are five squares with sides of length 2.
There is one square with sides of length 3.
There is one square with sides of length 4.
Total, 16.
I also corrected the bad grammar.