This notation is used when $Q \subseteq K$, where $K$ is some field and $x \in K$. Then $Q[x] = \{f(x) : f \text{ is a polynomial over } Q \}$, while $Q(x) = \{f(x) / g(x) : f,g \text{ are a polynomials over } Q \text{ and } g(x) \neq 0 \}$. If $x$ is algebraic over $Q$ then they are the same i.e. $Q[x] = Q(x)$. That's why you may be confused...

$\mathbf{Q}(x)$ is just the fraction field of the polynomial ring $\mathbf{Q}[x]$ over $\mathbf{Q}$.

$\mathbb{Q}[x]$ denotes the intersection of all $\textbf {rings}$ containing $\mathbb{Q}$ and $x$, while $\mathbb{Q}(x)$ denotes the intersection of all $\textbf{fields}$ containing $\mathbb{Q}$ and $x$.

If $x$ is algebraic over $\mathbb{Q}$, you can prove that $\mathbb{Q}[x]=\mathbb{Q}(x)$.