Well, I'm not entirely certain if you're asking whether they could agree on these things between civilizations, so that the Egyptians and Mesopotamians (for instance) could agree on what year it was, or whether within any of these cultural groups there was a commonly agreed calendar. In either case, though, the answer is broadly speaking "no".

Let's start from years. I'm most familiar with Mesopotamia, where the common practice was to give years names, based on some important event that took place that year, so you'd have a "year in which Shulgi, the strong man, the king of Ur, the king of the four quarters, smashed the heads of Urbilum, Simurrum, Lullubum and Karhar in a single campaign." These were then compiled into lists for reference, which to the historian's delight also serve to an extent as chronicles. In my understanding Egypt had a similar practice.

Of course, this system requires a year to be specifically named through, essentially, a royal decree, which means that a unified system of year names would only be in use in a single empire, if that. So even in Mesopotamia different states had their own lists that are not readily compatible, and of course the collapse of empires would cause havoc to these systems. Already ancient Mesopotamian scholars produced the Sumerian King List as a kind of composite that lists the regnal years of the dominant dynasty, though it starts from mythical rulers reigning thousands of years and, when dealing with historical rulers, occasionally appears to place contemporareous rulers erroneously in sequence. Essentially, keeping track of years accurately even within a single society was tricky, let alone aligning them with other empires.

I have no real expertise on calendars, but generally speaking the same issues apply. Most of the Near Eastern and Mediterranean cultures had lunisolar calendars, which require leap months to be added to align the years with seasons, and this was again done by decree, though presumably under scholarly recommendation. There was also no consensus on when a year started or what months were named, so again calendars would be local affairs unless the ruler of an empire was able to impose his own system of measurement on his subjects.

In closing I should add that all of this makes it quite difficult to accurately assign years, let alone dates, for ancient historical events. Though certain chains of years have been adequately established, there are very few ways (mainly astronomical phenomena mentioned in ancient sources) that would allow us to connect them to our calendar, and these allow for multiple interpretations, leading to multiple competing chronologies for the earliest historical periods.