There are many persecution elements in Matthew's gospel. What evidence is there that supports the idea that Matthew rhetorically used the reference to the deportation (RSV) in 1:11,12 and 17 to introduce the idea of persecution?
Any allusion to persecution in the repetition of ‘μετοικεσίας Βαβυλῶνος’ (‘deportation to Babylon’) in these three verses is extremely subtle, if it is present at all. The noun metoikesia appears only in these verses in the whole NT, and they offer no descriptive context to make a possible allusion explicit. The LXX uses the phrase similarly (per Thayer’s). The related word metoikizō appears in the NT twice: in Acts 7:4 for Abram’s ‘removal’ from Haran to Canaan, and in v.43 where Stephen (paraphrasing Amos) described the ‘removal’ to Babylonian as punishment for Judah’s idolatry.
This last usage is instructive, Acts being consistent with the theology of the Hebrew prophets:
The captives blamed the disaster of the Exile on their own impurity. They had betrayed Yahweh and allowed the Mosaic laws and cultic practices to become corrupt; the Babylonian Exile was proof of Yahweh's displeasure. During this period, Jewish leaders no longer spoke about a theology of judgment, but a theology of salvation. In texts such as Ezekiel and Isaiah, there is talk that the Israelites would be gathered together once more, their society and religion purified, and the unified Davidic kingdom be re-established.” ('The Hebrews: A Learning Module')
In later Jewish and Christian literature, Babylon became a potent symbol of pride, wickedness, and ultimate ruin. The Exile itself, however, was reflected on theologically not as ‘persecution of the righteous’ but as an act of God, a just punishment for an idolatrous people ... and as a turning point.
Matthew’s references, then, likely did not suggest persecution, but they were surely purposeful. Indeed, the repetitive 'deportation to Babylon' is the only historical marker in the entire timeline. It’s possible that Jechoniah – the ‘double personality’ (counting for two generations in v.11-12) – and his self-sacrificial exile to Babylon “established him as a forerunner of the Messiah” (Hood) in Matthew’s estimation. Or perhaps it was the connection to Cyrus, “YHWH’s messiah” (lauded in Is.45). Whatever his purpose in highlighting this “epoch-making event,” Matthew’s intention remains an open question among commentators. But there's no apparent suggestion here he intended to signal 'persecution' as a primary theme of his gospel.