
Genetically, blue eyes is a mutation believed to be from the Mediterranean area ~11,000 years ago. All blue eyed people descend from that mutation. The spreading of genetics, customs, cultures are all products of immigration. Even a place as genetically isolated as Ireland has genetic markers from Viking and Norman settlers. To be 100% of anything genetically implies more inbreeding than an Alabaman
The less genetic mixing in a population, the higher the probability that individuals inherit two copies of the same recessive allele, making recessive traits far more likely to appear. So no, blue eyes is not a “product of immigration,” at least not the modern definition of the word
The singular instance of mutation and early generations, yes. However, OP was most likely referring to the high frequency of blue-eyed people in Europe (70-95% across Northern Europe). The spreading of blue eyes and its prominence as a trait emerging in such a short time is a product of immigration.