I'm a native speaker of New Zealand English but as far as I'm aware this applies to any regional variant of English (although a "quarter" for 25¢ is specific to North America). A "quarter" is far more commonly used in conversation than a "fourth" but both mean "one part is four" (1/4). A "quarter" is a countable part so you can have two quarters or three quarters (such as pieces of a cake or pie). These could be separate quarter-sizes pieces or a fraction of the whole. However, a notable distinction is the there is a difference in the abbreviated or written forms commonly used in mathematics.
1/1 = 1 so it is a "whole" or "unit" (not a first)
1/2 (or 2/4) is "half" or "halves"
1/3 (or 1/3rd) and 2/3 is a "third" and "two thirds" respectively
3/4 and 3/4ths are read as "three quarters" or "three fourths" (these are not interchangeable)
Thus I would only use "fourths" in writing and mathematics, never in spoken conversation. Perhaps it is sometimes used by children or when teaching mathematics but is rarely spoken between two adults.
Notice that in these abbreviations the denominator (bottom number) corresponds to ordinal numbers: 1st "first", 2nd "second", 3rd "third", 4th "fourth", 5th "fifth", and so on. That is the convention for reading written fractions: x/y is read as x yths (not halves or quarters). The exception is "percentages": x/100 (x%) which uses the latin derived x per cent or very large numbers (such as ppm = parts per mil[lion]).