I’m not sure that this question can be answered with historical texts, and therefore I’m not really sure that it can be answered. I know that many of the oldest texts we have deal with measuring time. My background is in Hebrew linguistics, so I’m particularly familiar with things like the Gezer Calendar, but I also know that royal annals were very popular in the Ancient Near East. Many of these annals would say something like “and X became king in the 13th year of Y, he was 17 when he came to the throne.” But going back further than that, can we see any development of the concept of age, or have there been other cultures that used something other than years to measure age?
Followup question: was this sort of thing influenced by climate/biome? I'd imagine the presence or absence of cleanly delineable seasons would have an impact on an ancient culture's concern with measuring years before the adoption of agriculture.