Is the phrase "a high number of" considered correct? Or is it only correct to say "a large number of"?
Japan has a high number of active volcanoes.
I consider it awkward in the context of "Japan has a high number of active volcanoes", but it's frequently used that way in biomedical articles. Saying "a large number of" or "many", in this case, is much better style, IMHO, but it's not ungrammatical.
Style is rarely judged as "correct" or "incorrect", except in contests for which a certain style has been prescribed and others have been proscribed. Writing style is usually judged on a sliding scale that runs from Godawful! (-100) to Sublime! (+100), depending upon one's taste. OTOH, style manuals may deal with this kind of structure: that's case by case.
I always change "a high number" to something more appropriate when the quantity or amount is what's important rather than the size of the number: Sometimes a great number?.
I would suggest that using "high" or "low" might in this case suggest a relative quantity, whereas "large" or "small" might be more absolute.
So four volcanoes could be regarded as a high number of volcanoes because most other countries have fewer, but to describe four as a large number of volcanoes could sound awkward because objectively the reader would not think of four as a large number.