Assume $K_\min \le K \le K_\max$, and introduce binary decision variables $x_i$ for the three cases (reordered as 1, 3, 2), together with linear constraints
\begin{align}
\sum_{i=1}^3 x_i &= 1 \tag1\label1 \\
K_\min x_1 + Q_\min x_2 + Q_\max x_3 \le K &\le Q_\min x_1 + Q_\max x_2 + K_\max x_3 \tag2\label2 \\
Q_\min \le Q &\le Q_\min x_1 + Q_\max(1-x_1) \tag3\label3 \\
(Q_\min-K_\max)(1-x_2) \le Q - K &\le (Q_\max-K_\min)(1-x_2) \tag4\label4 \\
Q_\max x_3 + Q_\min(1-x_3) \le Q &\le Q_\max \tag5\label5
\end{align}
Constraint \eqref{1} selects exactly one case.
Constraint \eqref{2} enforces the bounds on $K$ for each case.
Constraints \eqref{3} through \eqref{5} enforce the value of $Q$ for each case.
You could alternatively replace \eqref{3} through \eqref{5} with indicator constraints:
\begin{align}
x_1 = 1 &\implies Q = Q_\min \\
x_2 = 1 &\implies Q = K \\
x_3 = 1 &\implies Q = Q_\max
\end{align}
Yet another approach is to recognize that $Q$ is a nonconvex and nonconcave continuous piecewise linear function of $K$ and use any formulation for that, such as the ones recommended in the answers to How to linearize specific range constraints?.