For a random variable $X$ with cumulative distribution function $F(x) = P(X\leq x)$, the usual definition of the quantile function is
$$
Q(p) = \inf\{x: F(x)\geq p\},\quad p\in(0,1)
.$$
Now if $X$ is absolutely continuous, then $F$ is continuous, monotone increasing and, in this case, $Q = F^{-1}$.
Now, in the continuous case, for any $p\in (0,1)$, the $p$th quantile, say $\xi_p = Q(p)$, can be seen as
the particular value of the r.v. $X$ which leaves probability mass $p$
on the left and probability mass $1-p$ on the right.
In other words, the $p$th quantile $\xi_p$ is s.t.
$$
p = P(X\leq \xi_{p}) = F(\xi_{p}).
$$
This interpretation holds for any distribution, the beta included.
For the discrete case consider the following example. Let $X$ be a discrete random variable with probability density function
$$f(x) =
\begin{cases}
1/3 &\text{if } x=-1\\
1/6 &\text{if } x=0\\
1/2 &\text{if } x=1\\
0 &\text{otherwise}
\end{cases}.
$$
The distribution function is
$$F(x) =
\begin{cases}
0 &\text{if } x \in (-\infty, -1)\\
1/3 &\text{if } x \in [-1, 0)\\
3/6 &\text{if } x \in [0, 1)\\
1 &\text{if } x \in [1, \infty)
\end{cases},
$$
and the quantile function is
$$
Q(p)=
\begin{cases}
-1 & \text{if } p\in(0,1/3]\\
0 & \text{if } p\in(1/3,3/6]\\
1 & \text{if } p\in(3/6,1)\\
\end{cases}.
$$
You can if you wish set $Q(0) = -\infty$ and $Q(1) = \infty$.
