Learn English – Is “more quickly” grammatically correct?

Yes, that works.

You can also say

• I can move quicker than she can.
• She moves quickly, but I can move quicker still.
• Between us, I move quickest.
• Between us, I move quicker.
• I am even quicker than she.

Edited to Add: Since I got a complaint and a downvote for using "quicker" and "quickest" as adverbs (when they seem nominally to be adjectives), I thought I would put that as a question, and garnered responses that seemed to militate strongly towards my usage here to be legitimate. A paraphrased "Casual yet still grammatical" was the judgment of one commenter who consulted an online dictionary about adverbial use of a nominal adjective.

So I will stand by my answer, and invite the downvoter to reconsider.

Edited further to Add: I had never heard of the term "flat adverb" before, but @FumbleFingers directed my attention to it on grammarist.com:

"Some adverbs don’t change from their adjectival forms. These are known as flat adverbs. A few of the most common ones are close, deep, fast, quick, and right. Some of these have corresponding -ly adverbs with which they are interchangeable—for example, come quick and come quickly mean the same thing."