Calculation of probability. Why is my solution wrong?

Others have already commented on why the solution you pose overcounts. Instead, I offer an alternate methodology that does not use Inclusion/Exclusion, but instead focuses on the addition and multiplication principles. First, I separate the problem into disjoint cases, then I count the number of elements in each disjoint case:

Let there be $x_1$ winter birthdays, $x_2$ spring birthdays, $x_3$ summer birthdays, and $x_4$ fall birthdays. We have:

$$x_1+x_2+x_3+x_4 = 7, \forall i, 1\le x_i$$

Each solution to this equation corresponds to a solution to the equation:

$$y_1+y_2+y_3+y_4 = 3, \forall i, 0\le y_i$$

(The bijection between solutions is $y_i = x_i-1$).

For each solution to this equation, consider a multiset:

$$\{\text{Winter}\cdot (1+y_1), \text{Spring}\cdot (1+y_2), \text{Summer}\cdot (1+y_3), \text{Fall}\cdot (1+y_4)\}$$

Each permutation of one of these multisets corresponds to a distinct possible outcome of birthdays where all four seasons are represented. It is not difficult to show that as you span all possible solutions to the Diophantine equation and all possible multiset permutations, you find all possible ways to assign the birthdays. So, the next step would be counting them all.

You can have: 3,0,0,0 (in some order) 2,1,0,0 (in some order) 1,1,1,0 (in some order)

This gives three cases: 3,0,0,0, choose one season to have 4 birthdays while every other one has 1, then permute the multiset: $$\dbinom{4}{1}\dfrac{7!}{4!1!1!1!}$$

Case 2: 2,1,0,0 Permute the multiplicities, then permute the multiset: $$\dfrac{4!}{2!1!1!}\cdot \dfrac{7!}{3!2!1!1!}$$

Case 3: 1,1,1,0 $$\dbinom{4}{3}\dfrac{7!}{2!2!2!1!}$$

Add all of these up and divide by $4^7$ and you should get the same answer as the book.

$$\dfrac{8,400}{16,384} = \dfrac{525}{1,024} \approx 51.27\%$$