Monthly: once a month
Quarterly: once every three months
???: once every four months
Semiannually: once every six months
Seasonal is an approximation.
Thrice-yearly.
If you take bimonthly to mean once every two months, perhaps bi-bimonthly?
Quadrimestrical
or quadrimestral
triannual
Thanks. It looks like the short answer is “yes,” but it’s definitely not a common word, not appearing in the “regular” m-w.com.
well
there is that very rare and unused business term…quarterly 
I’ve also heard that some ivory tower institutions use something similar.
But as the OP noted, “quarterly” means “once a QUARTER”, i.e., every three months, not every four months.
ETA: I don’t mean to sound snarky but I’m genuinely curious: did you really not know the difference between “quarterly” and “every four months”?
might have meant something there. Of course I went to a school that operated on a semester system.
It was a bad joke.
But that means once every trimester, i.e., four times each year or three times in the average human pregnancy.
Does triannual have to be every four months - or does it just mean 3/year? Obviously anything that occurs every 4 months will occur 3 times per year, but not everything that occurs 3 times per year necessarily occurs every 4 months.
Take for instance, something that occurs on April 5, June 16 and December 8 every year. Could such a thing be called a triannual occurence? If so, then triannual wouldn’t fully describe something that occurs once every 4 months (although it wouldn’t be incorrect to call such a thing triannual). If you can’t call such a thing triannual, then what would you call it?
Heptadeciweekly
But the same issue arises with any such word. Yearly, monthly, weekly, etc. Yearly doesn’t necessarily mean that something happens on exactly the same day each year. I’m about to have my yearly visit to our UK office. Last year it was in September.
The “deci” prefix means “one tenth”. I think you meant “heptadekaweekly”.