Count them?

For 18 1 row, there will be 1+2+3+...+18 = 171 rectangles. If such rows are repeated 19 times, the multiplicity of the same in other direction is: 1+2+3+.....+19 = 190. Total number is: 171 190 = 32490.

We can view a $18\times19$ grid as a grid formed by 19 horizontal lines and 20 vertical lines. A rectangle in the grid can be formed by using 2 horizontal lines and 2 vertical lines. Hence, there are $\binom{19}{2}$ of choosing a "width" of a rectangle and $\binom{20}{2}$ ways of choosing a "length" of a rectangle. Thus the total number of rectangles is $\binom{19}{2}\times\binom{20}{2}=32490$ .