Sorry for taking so long to respond. I really had to consider my answer and how to approach it. As far as the modern day sense of colonization I will firmly say no. As for the second, I lean toward yes, with the caveats that ancient colonization isn't my area of expertise and it seems what qualified as a colony is a bit ambiguous. Also, I can only speak about the Bronze Age vis a vis the Ancient Near East.

First, we must consider how the civilizations of the Bronze Age operated. It is difficult for us to characterize any of the Bronze Age civilizations as empires (a precondition for any sort of modern day colonization) until after the Late Bronze Age collapse of the 12th century BCE. For most of the Bronze Age, various kingdoms such as the Babylonians, Assyrians, Hittites, Egyptians, Elamites or the Mitanni didn't really impose their will territorially over different groups of people from a central authority and therefore cannot be considered as empires. The modus operandi of Bronze Age kings was to form up during the fighting season with a sizeable army and raid and pillage nearby kingdoms. These armies wouldn't occupy any territory and as soon as the king had accomplished his goals and destroyed a few cities, they'd return home. I can give one famous examples of this practice. The 1595 BCE sack of Babylon by Hittite king Mursili I shows just how far a king would go to inflict defeat on his enemy. His army traveled over 1000 miles from Anatolia into Mesopotamia, looted and destroyed Babylon, then just as soon as he 'd arrived he returned home. Mursili had no notions of actually conquering and occupying Babylonia or Assyria.

Since these Bronze Age kingdoms couldn't be thought of as Empires in even the ancient sense of the word, we can rule out any sort of imperial colonization or exploitation such as that engaged by the British or French in later times. Conquering armies took loot, slaves, and burned cities to the ground, but Bronze Age kingdoms had no desire to create empires until after the Bronze Age collapse. Afterward, we begin to see what we would recognize as imperial ambitions by the Neo-Assyrians. But this doesn't begin to happen until about the 9th century BCE. Ancient kingdoms were sometimes brutal towards one another, but ancient warfare usually meant simply showing up to a city, destroying it, and then promptly returning home.

The second idea is that of a spreading out of peoples like the Greeks or Phoenicians. I'll admit this question is harder for me to answer as it isn't really my area of expertise (if I have any expertise at all!). I believe that the Mycenaean Greeks colonized the islands of the Aegean Sea. The Minoan Civilization of Crete made it as far as Rhodes, founding a colony there in the 16th century BCE. There is some evidence of Phoenicians beginning to spread during the 15th century as well. So I would say yes, the spreading out of peoples was underway during the Late Bronze Age. What I see more of in terms of "colonization" is the trading colony, something which was widespread during the Bronze Age. Old Assyrian traders were known to have occupied a portion of the Hittite city of Kanesh during the 18th century BCE. While not a colony in the traditional sense of your question, these people lived and worked for generations in these cities. This was a family affair, with the husband living in Kanesh, organizing trades between his outpost and the home city in Assyria. We have correspondence between husbands and their wives who would control the family business in their absence. I know that this type of "colonization" was common in the Ancient Near East.

Sorry I couldn't give a more definitive answer, but as with most ancient history the evidence is ambiguous and many different theories exist.

In an effort to answer this question I’d like to clarify. Do you mean “colony” in the modern sense? As in moving in from long distance and displacing/exploiting a native population? Or are we referring to a people spreading from a central location into other areas like the Phoenicians?

The great thing about the Bronze Age, at least from the point of view of ancient Mesopotamia, is that there are numerous records in the form of cuneiform tablets from which to work. Our knowledge of Ancient Assyria and Babylonia is considerable considering how long ago the events of this period took place.

I highly recommend Eric C. Clines 1177 B.C. The year Civilization Collapsed. This book chronicles the collapse of the Late Bronze Age and gives a good overview of all the major players of the Ancient Near East, including the Mycaneans and Minoan Greeks. I cannot recommend this book enough as it is both academic (in my opinion) but also accessible to the everyday reader.

Frances Joannes The Age of Empires chronicles a significant portion of Ancient Near East history.

For something more along the lines of an academic textbook that will give you extensive general knowledge of the time period, Marc Van De Mieroop's A History of the Ancient Near East is really unrivaled, in my opinion. It's easily accessible to a reader and it also starts all the way back from the end of the Neolithic period up until the fall of Babylon to the Perisans and beyond. Really a great read.

Good luck and I hope this helps!