A trimester is a period of three months (a quarter of a year or one third of a pregnancy) http://oxforddictionaries.com/definition/english/trimester
Ahh. Forgive me. Your math is certainly correct, and your explanation cleared things up for me. I think my confusion stems from how we use the words quarters and trimesters in the US, which I assume is slightly different than UK English. Quarter obviously refers to anything in fourths, with the Quarter coin (aka, 25 US cents) being the most common use of the word in US English, but the word trimester is almost exclusively used to describe the rare trimester term system in some universities, which breaks the school year into 1/3's, but most commonly, and as you already noted, the terms of pregnancy. Apart from the seasons, we rarely think about the fact that a trimester is also 1/4 of the year, which as you already noted is a quarter of the year.
The real question is what on earth Duolingo's sentence means. If a trimester is defined as a period of three months there doesn't seem much chance that it could be any shorter :-) So I guess they really do mean it in your sense, i.e. the length of a term at a university which may vary between institutions - some shorter, some longer.
Ohio State Univ. used "quarters" when I first started and there were 3 quarters during the 9 month normal school year plus a two part summer session (which would make up the fourth quarter. Winter tended to be the short quarter by having less days in school. Most other schools had a two semester system during the regular 9 month school year, and a separate summer course time. -- a bit of history