If I want to say the first three months of the year in one word, I would say "the first quarter".
Which word should I use to refer to a period of four months?
It isn't common to speak of a four-month period, at least in the United States where I live. Thus, although others have suggested words for this, my recommendation is to not use them. Very few people would use such a word in English; the more natural way is to speak of a four-month period or some variation on that expression.
Sales were down during the four-month period ending (or beginning) in April.