Expressing a constraint of the form $\max(x_1,x_2) \ge q$ in a linear program
No. If you could do that in linear programming then you could force a variable to have binary values, so you'd be able to solve integer linear programs using LP solvers.
Indeed, we can simulate $x \in \{0,1\}$ with:
$$
\max\{x, 1-x\} \ge 1, \\
x \ge 0, \\
x \le 1.
$$