It means “in a row”.

• Those scholarship are granted up to three quarters in a row.

It might also mean “in one single attribution”.

• Those scholarships are granted by a maximum of three quarters in a single attribution.

Based on the Larousse.fr definition of “trimestre” quoted below, I think that “trois trimestres” in your context is referring to the three, 3-month-long terms/divisions of a typical nine-month-long French academic year:

[trimestre =] Chacune des trois divisions de l’année scolaire française
(de septembre à juillet) équivalant approximativement à trois mois et
séparée par des vacances.

With this interpretation of “trimestre” in mind, “jusqu’à trois trimestres à la fois” could be replaced with “jusqu’à une année [scolaire] à la fois” in your sentence, which I believe could be rendered in English as “for up to [a maximum of] one [school/academic] year at a time” as it is used to describe these nursing scholarships:

[These] Scholarships are awarded for up to [a maximum of] one year at a time [with the possibility of renewal].