Is there a proper term to describe ⅓ of a year (4 months)?
Tertile.
The term for a four month period is quadrimester. Quad = 4 mense= month
There is a difference between a duration period of three months as in "trimester" and an event occurring every 3 months as in "quarterly". In the same vein you would have a quadrimester or tri-annually(which means three times a year; not every 3 months) both being correct in the right context.