Difference between "flip through" and "flick through"
Solution 1:
As the comments indicated, "flicked through" appears to be common in British English, although they seem to use "flipped through" almost as often. Here's an Ngram with the "British English (2009)" corpus:
On the other hand, American English doesn't use "flicked through" as much. Here's the graph using the "American English (2009)" corpus:
Beyond that, a writer may alternate between flicking and flipping through pages for some variety. Both words are onomatopoeic and they create slightly different sounds in my mind as I read them.