Is it "walk quickly" or "quickly walk?"
What is the correct way to phrase the sentence:
Please walk quickly.
Please quickly walk.
Thank you.
To do something quickly is to do something with speed. To quickly do something is (normally) to do something very soon.
Hence, for the instruction in your example, it must be Please walk quickly.
To extend the explanation with further examples, these two sentences have different meanings:
He put down his pen and walked quickly over to my desk.
He put down his pen and quickly walked over to my desk.
In the first sentence, his walking is at speed. The second sentence is ambiguous: it could mean that he started walking very soon after putting down his pen, or it could mean that the total duration of the walk was very short.
They are both grammatically correct (technically), but "Please walk quickly" is by far more common.
This is backed up with ngram evidence here.
Verb/adverb order is tricky, and fully of idiosyncracies. You might enjoy reading through the answers in this Stack Exchange question.
That's not to say that there aren't rules -- just that there are a lot of rules and, even then, many of them depend on what the reader wants to empahsize.
There are also lots of more details examples if you search for "adverb verb order" in your favorite search engine.
According to Dictionary.com
quickly [kwik-lee]
adverb
1. with speed; rapidly; very soon.
It depends on how you use these words and what word you use. They are both considered grammatically correct. If you are using the word walk it is more common to use quickly after walk.
Here are some example sentences from Your Dictionary.com
~The sting of his cool hospitality was quickly replaced with awe as she turned back to the room.
~I quickly learned that each printed word stood for an object, an act, or a quality.
~The storm passed quickly, but the night remained warm.
~They marched very quickly, without resting, and halted only when the sun began to set.