Is "tri-quarterly" a real English word meaning 3 times a year? Are there any other words that mean 3 times a year?
If "tri-quarterly" means anything, surely it would be every third quarter (every nine months). Or possibly three times a quarter, which is monthly. You can't redefine a quarter as a third, though.
Three times a year is triannual — not triennial which is every three years. You could also say every four months; "every four months" is preferable because it removes the possibility of confusion between triennial and triannual.