Is there a way to turn down my PS4 controller's speaker volume?

Try this:

1. Hold the Playstation Button

2. Select Adjust Devices

3. Lower the volume

_(Source)

You can set the controller speaker volume in the console's system settings:

Settings > Devices > Controllers > Volume Control (Speaker for Controller)

From there, you can adjust the volume, which will apply to all games.

Edit: Timmy Jim's answer provides a much quicker and simpler way of doing this.