No, you have not damaged anything. While I do it all the time because it saves time to power off and back on, it is best to turn the camera off to reduce the probability of dust adhering to the sensor.

When the camera is on, the sensor is usually powered which creates a static charge. This makes it more likely for light dust to stick to the sensor but that does not mean you will not get any dust when you change the lens with the camera powered off.

Just try to make the change quickly and have the camera point downwards as much as possible. This will reduce the chances of dust going into the sensor chamber.

The year at Photo Plus Expo, held at the end of October, Sony repeated this many times during their presentations, so this information is very recent. It is unlikely that this is different for any other brand, considering that Sony makes the sensors for most brands except Canon.