You're probably thinking of The Final Reflection by John M. Ford.
I bought it in the mid-1980s, when it was a newly released paperback, and I loved it. But I believe a lot of the details about Klingon culture and biology which Ford invented to flesh out his characters were then firmly ignored by lots of other writers who contributed new material to Star Trek in one way or another before the recent reboot -- people who wrote novels, TV scripts, movie screenplays, etc. -- and that's probably just as well.
I remember it was made clear within the text that Klingons matured significantly faster, and died of old age (if they lived long enough) much younger, than did the humans of that era. (Note: This was set decades before James T. Kirk became captain of Enterprise. A very young Spock appears onstage in one scene -- the one depicted on the cover -- and little Leonard McCoy is stated (by his grandfather) to still be in diapers.) I don't remember if we were told exactly at what age the average Klingon reached puberty, and/or became regarded as a "legal adult" by the laws of their Empire. It's been a long time since I last reread my copy of the novel.
With that said, I'll acknowledge that "reaching puberty around age eight" is much closer to what I vaguely remember than "might drop dead at age eight" would be. I believe there was one point where a major human character -- Dr. Tagore, appointed to be a Federation Ambassador to the Klingon Empire (the first such Ambassador, I think) -- asks Captain Krenn (the Klingon who is the main viewpoint character) about his age, is told that Krenn is about twenty-five (or a little under) and says something to the effect that one way to look at it was that both of them might last another twenty or thirty years, if nothing violent happened to cut their lifespans short. (Dr. Tagore was, I think, somewhere around age 70 at that point.) Sorry I can't look it up for you to get the exact wording, but my copy must be inside a box in a storage unit at the moment.