No, that is not true at all. The first inhabitants of Ireland would have been Mesolithic (or possibly even Paleolithic - though the only definitive graves discovered thus far date to the Late Mesolithic) hunter-gatherers between 7,000 and 6,000 BC via a land-bridge that connected the island to continental Europe. This occurred long after the Pleistocene Ice Age, which ended around 11,700 years ago. During the Ice Age itself, only a small portion of southern Ireland would have not been covered by glaciers and by the time of definitive human colonization, .

The next wave of settlers arrived around 4,000 BC - these were neolithic farmers who would have found a lush and heavily forested landscape that had barely been altered by their Mesolithic hunter-gatherer predecessors. They would have settled in higher altitude areas and cultivated cereal crops, pursuing deforestation to claim land for agriculture, wood for building, fuels for fires etc. The Neolithic migration was really the last major influx of immigrants - the Irish genetic makeup is overwhelmingly dominated by them with Central European (Celts) making only a marginal contribution.

This is because eras like the Bronze and Iron Ages were likely "imported" to Ireland rather than the result of massive conquests and colonization. In fact, it is probable that "Celtic" material culture and language were imported through trade and diplomacy with Celtic speakers living in what's now France or Spain sometime around 500 BC. Local Irish elites probably imported prestige goods from the Continent and Britain and might have adopted their Goidelic language as a sign of prestige and a more efficient means of communication. It's problematic to think of the Irish as inherently "Celtic" because that word only really refers to the Celtic languages, which Irish people have only been speaking for some 2,500 years, when Ireland was first colonized almost 9,000 years ago!