Not a teacher
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Hello, tufguy!
The problem with your sentence is that you're trying to tell someone to modify how they're speaking when they've already started speaking. If you want to use an adverb, you should make it a comparative. In other words: [STRIKE]quietly[/STRIKE] → more quietly.
The person is already speaking in a certain way, in this case, loudly. You can think of the way they're speaking as a point of reference, and the way you'd like them to speak is either louder or quieter than that. That's why we use comparatives in such situations. It's similar to how a learner might want to say "Could you speak more slowly?" when the person is speaking too fast for them to follow.
The sentence "Please, speak quietly" is not ungrammatical. It just doesn't fit the situation you want to use it in. It does, however, work perfectly fine if you want to advise someone how they should speak before they start speaking. Suppose you want to introduce your significant other to your grandma for the first time. Being as old as she is, she has a hearing problem, and being as forgetful as she is, she often forgets to put on her hearing aid. You might say "You need to speak loudly to her". Your partner has a certain idea of how loudly they should speak, but that's entirely subjective and it may not be loud enough. After they've spoken a few times, but your grandma didn't reply, you might want to say to your significant other "You need to speak more loudly; she can't hear you".
Since this thread has ventured to the topic of politeness, rudeness, and abnormality of certain cultures, I'd like to point out that being rude is not incorrect. It may be inadvisable, but it certainly has its uses and is sometimes well called for. When dealing with certain people, you might want to use something rude and nasty to get your point across and achieve the effect you want, especially if all the polite linguistic tools have proven ineffective. A simple shut up, optionally with a bit of profanity in between, may be rude, but sometimes it's what you need to resort to to get the job done.