For an airplane, this occurs at the maximum speed at minimum altitude corner of the flight envelope.
For many vehicles, if launched at full throttle, the aerodynamic forces would be higher than what they can withstand.
For this reason, they are often throttled down before approaching max q and back up afterwards, so as to reduce the speed and hence the maximum dynamic pressure encountered along the flight.
[2] The three Space Shuttle Main Engines were throttled back to about 65–72% of their rated thrust (depending on payload) as the dynamic pressure approached max q.
[3] Combined with the propellant grain design of the solid rocket boosters, which reduced the thrust at max q by one third after 50 seconds of burn, the total stresses on the vehicle were kept to a safe level.