Yes, the phrase ‘fairly quickly’ is considered proper grammar. It consists of an adverb ‘quickly’ that describes how something is done, and ‘fairly,’ which functions as an adverb of degree. This means that something is done at a speed that is more than average but not extremely fast.
In everyday conversation and writing, using ‘fairly quickly’ is common and understood by most English speakers. It’s an appropriate way to convey a moderate level of speed without being too extreme. For example, you might say, ‘She completed the project fairly quickly,’ indicating that while she didn’t rush, she was efficient in her work.