Teaching the speak command allows you to teach the quiet command more easily.
HOWEVER, you can teach quiet by simply rewarding the behavior you want. If the dog is barking, you wait.. click/say yes and reward the instant the dog is silent (I am assuming you use a clicker or the word "yes!" to mark the correct behavior and the dog understands that the marker is followed by a reward). if the dog is in drive such as running and playing excitedly, this is less effective since puppy play is self rewarding.
IF you can get the dog to offer silence for the marker and reward, you add a "Quiet" command cue. Say it low and long as the tone will add to the calming effect.
Maybe you know this.. but shouting at a dog that is barking escalates the barking. Dogs just hear your tone. They are keyed up and barking and now (to them) shouting sounds like you are "joining in."
I have a few issues with teaching "Quiet" to stop barking.
1) not all barking is intentional - it can also be an emotional response that's not necessarily under the dog's control - like laughing or crying in humans. In order to stop the barking in this instance, you have to identify and change (or at least reduce) the underlying emotional response.
2) dogs are totally capable of learning that what they are getting rewarded for is stopping barking - which requires starting to bark in the first place.
But, if you do want to teach a quiet, the way to do it is to let the dog bark, and then say "yes!" and give him a cookie for stopping. Eventually the dog will start to expect a cookie when he stops barking and will start to look at you, and you can start to put a hand signal on it. Simple in principle, much more difficult in practice. Easier to teach the dog not to bark in the first place, IME.
Instead of "quiet", I taught my dog an "It's OK" command which I use in any situation he is agitated to calm him. It doesn't work every time, but I also use it in situations when he is nervous like during fireworks or thunderstorms and it seems effective. He even takes it as a cue to cuddle up and tries to climb in my lap. I made the mistake of saying that while giving him a bath last night and ended up with a wet dog climbing on me lol. It even works during zoomies if I can get his attention or when he panics like if he gets tangled in his leash. He tends to ignore it when he is excited though, so I have been working on a quiet command too. I taught it by saying "It's OK" while rubbing his chest and giving him cuddles. It didn't take long before he naturally seemed to pick it up by association. I have no idea if this would work for other dogs and if it will continue to work, or if I just got lucky by teaching it inadvertently.
I agree with allowing release and expression... My guys going into this howling thing.. they not nice vocal like the beauty harmony of a group of German Shepherds. Sounds more like I am beating a group of baby seals to death.. I do smile and let them know how sweet it was lol...