Четверть means a quarter of anything. Четверть яблока (though in this case четвертинка would be more appropriate), четверть дня, четверть пути, etc. It means "term" in a school only when there are four terms, which is very standard in Russian secondary schools.
If there are three terms, you can't call them "четверть". The best choice here would be "триместр". However, it's very very very uncommon for Russian secondary schools to have three terms, so the term "триместр" is by far not as well-established and widely known as "четверть".
Three terms are more common for universities, not secondary schools (although even there, three terms are very uncommon). The standard situation in universities is to have two terms called "семестр", and I guess that's why for three terms we use "триместр". I suppose that if three terms were more common in secondary schools than in universities, then they would probably have been called "треть", not "триместр".
(Also note that "семестр" and "полугодие" differ specifically in that the former is used in universities and the latter, in secondary schools.)