A bunch of lilac consists of flowers with 4 or 5 petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with 4 petals be divisible by the number of flowers with 5 petals?
1. Let \( q \) be the number of flowers with 4 petals and \( c \) be the number of flowers with 5 petals. According to the problem, both the total number of flowers and the total number of petals are perfect squares. Therefore, we can write:
   \[
   q + c = p_2^2
   \]
   and
   \[
   4q + 5c = p_1^2
   \]
   where \( p_1 \) and \( p_2 \) are integers.

2. Assume, by way of contradiction, that the number of flowers with 4 petals is divisible by the number of flowers with 5 petals. That is, let \( q = cx \) for some integer \( x \).

3. Substitute \( q = cx \) into the equations:
   \[
   cx + c = p_2^2 \implies c(x + 1) = p_2^2
   \]
   and
   \[
   4(cx) + 5c = p_1^2 \implies c(4x + 5) = p_1^2
   \]

4. From the first equation, we have:
   \[
   c(x + 1) = p_2^2 \implies x + 1 = \frac{p_2^2}{c}
   \]
   Let \( y_2 = \frac{p_2^2}{c} \), then:
   \[
   x + 1 = y_2 \implies x = y_2 - 1
   \]

5. Substitute \( x = y_2 - 1 \) into the second equation:
   \[
   c(4(y_2 - 1) + 5) = p_1^2 \implies c(4y_2 - 4 + 5) = p_1^2 \implies c(4y_2 + 1) = p_1^2
   \]
   Let \( y_1 = 4y_2 + 1 \), then:
   \[
   c y_1 = p_1^2
   \]

6. From the equations \( c y_1 = p_1^2 \) and \( c y_2 = p_2^2 \), we see that \( c \) must divide both \( p_1^2 \) and \( p_2^2 \). Since \( p_1 \) and \( p_2 \) are integers, \( c \) must be a perfect square.

7. Consider the equation \( c y_2 = p_2^2 \). Since \( c \) is a perfect square, let \( c = k^2 \) for some integer \( k \). Then:
   \[
   k^2 y_2 = p_2^2 \implies y_2 = \frac{p_2^2}{k^2} = \left(\frac{p_2}{k}\right)^2
   \]
   Since \( y_2 \) is an integer, \( \frac{p_2}{k} \) must also be an integer. Let \( \frac{p_2}{k} = m \), then \( p_2 = km \) and \( y_2 = m^2 \).

8. Substitute \( y_2 = m^2 \) into \( y_1 = 4y_2 + 1 \):
   \[
   y_1 = 4m^2 + 1
   \]
   Therefore:
   \[
   c y_1 = k^2 (4m^2 + 1) = p_1^2
   \]

9. Since \( p_1^2 = k^2 (4m^2 + 1) \), \( 4m^2 + 1 \) must be a perfect square. Let \( 4m^2 + 1 = n^2 \) for some integer \( n \). Then:
   \[
   n^2 - 4m^2 = 1
   \]
   This is a Diophantine equation known as the Pell's equation. The fundamental solution to this equation is \( (n, m) = (3, 1) \), but this does not satisfy our original problem constraints.

10. Therefore, no integer solutions exist for \( c \) and \( q \) such that \( q = cx \) and both \( q + c \) and \( 4q + 5c \) are perfect squares.

Conclusion:
\[
\boxed{\text{No}}
\]