On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square?
1. **Identify the coordinates of the opposite vertices of the square:**
   The given points are \((1, 2)\) and \((7, 4)\).

2. **Calculate the length of the diagonal of the square:**
   The length of the diagonal can be found using the distance formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Substituting the given points \((1, 2)\) and \((7, 4)\):
   \[
   d = \sqrt{(7 - 1)^2 + (4 - 2)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40}
   \]

3. **Relate the diagonal to the side length of the square:**
   For a square with side length \(s\), the diagonal \(d\) is given by:
   \[
   d = s\sqrt{2}
   \]
   Given that the diagonal \(d = \sqrt{40}\), we can set up the equation:
   \[
   \sqrt{40} = s\sqrt{2}
   \]

4. **Solve for the side length \(s\):**
   \[
   s = \frac{\sqrt{40}}{\sqrt{2}} = \sqrt{\frac{40}{2}} = \sqrt{20}
   \]

5. **Calculate the area of the square:**
   The area \(A\) of a square with side length \(s\) is given by:
   \[
   A = s^2
   \]
   Substituting \(s = \sqrt{20}\):
   \[
   A = (\sqrt{20})^2 = 20
   \]

Conclusion:
\[
\boxed{20}
\]