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.
I'm not sure I've seen a word searched so much and never found due to nonexistence. I'm searching too and found this post among others (which go back 15+ years). My context is an Income Statement for a firm (the sales example above is helpful in this regard). So I need a "label" that describes a 4-month period, or a unit of time that is distinct from other 4-month units of time. For a monthly income statement column heading, we simply say "Month: 1, 2, 3, 4, 5, ....or Jan, Feb, Mar, Apr...". For quarters we simply say "Quarter: 1, 2, 3, 4". But a business whose projects have a distinct 4-month lifespan, thereby generating 3 sequential spans per fiscal year (in my case a fiscal year is Jan through Dec U.S. tax year), or equal 4-month periods. I'm going with a statement heading of "Quad-monthly Income Statement" and my 4-month periods will be labeled "Period: 1, 2, 3".
Your question seems self contradictory. You say, "'If I want to say the first three months of the year in one word, I would say "the first quarter".' Except, umm, "the first quarter" is three words and not one word.
The short answer to your question is: There is no commonly used word in English for a period of 4 months. I see some people have posted suggestions for an invented word that would mean this, but if you tried to use a made-up word, you would have to explain it or no one would know what you were talking about.
If you are just referring to this time period one or two times in some discussion, it would be far simpler to just say "a period of 4 months". Yes, that's 5 words instead of 1, but if you made up a new word, it would take more than 4 words to explain it, so you'd end up using more words, not less, not to mention being more confusing.
If you are writing a long article or a book in which you will repeatedly refer to a period of 4 months, then it could be useful to invent a word. If you use it often enough, the "overhead" of having to define it is spread across many uses and it may be worth the trouble.
There isn't a word in common use for a period of 4 months. But why would you need another word when you have the word "month"?
You asked if there was "a single word" - well, "month" is a single word, you just need to pluralise it and put the number four in front of it to show how many you mean.
Think about it - if there was a single word that meant "four months", you'd need to state how many of them you meant anyway, if not with a number then with the indefinite article "a". For example, in British English we do have a word meaning two weeks - we call it a 'fortnight'. You can say "a fortnight" or "two weeks" - both require using two words.
Words like "trimester" and "quarter" which mean three months are not really evidence that we should have a word for every quantity of months we might need to refer to. They are for specific things. For example, a 'quarter' of a year is always January-March, April-June, July-September, or October-December. It doesn't matter whether you are referring to a calendar year or a financial year, those are the quarters. You can't refer to the 3 months of Febuary - April as "a quarter". Likewise 'trimester' refers to three months of a specific term, such as a pregnancy or a school year. More importantly, a pregnancy trimester is 13 weeks, not 3 calendar months.
In cases where there are no words [yet] but you need one, it is not a stretch to invent one. People do invent words when they need one, even in "official" publications.
Of course, words are used in multiple contexts, so you first need to have a context in mind. Eg in writing:
Month: 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th.
Quarter: 1st, 2nd, 3rd, 4th.
..I will complete the pattern with any of the following:
Tri-ter: 1st, 2nd, 3rd.
Thirter: 1st, 2nd, 3rd. //
Thirder: 1st, 2nd, 3rd.
Thirdth: 1st, 2nd, 3rd.
Thirth: 1st, 2nd, 3rd. //
Eg 2 in writing:
I need it for half a year.
I need it for a quarter year.
..I will complete the pattern with any of:
I need it for thirth a year. //
I need it for a thirter year. //
I need it for a third a year.
I need it for one-third a year.
I need it for onethird a year.
I need it for 'third a year.
I need it for thirdth a year.
And if one can consider the candidates that are usable in multiple contexts to be superior.. Thus "thirth" or "thirter".