[Luke 19:44 NLT] (44) They will crush you into the ground, and your children with you. Your enemies will not leave a single stone in place, because you did not accept your opportunity for salvation."
[Luke 19:44 MGNT] (44) καὶ ἐδαφιοῦσίν σε καὶ τὰ τέκνα σου ἐν σοί καὶ οὐκ ἀφήσουσιν λίθον ἐπὶ λίθον ἐν σοί ἀνθ’ ὧν οὐκ ἔγνως τὸν καιρὸν τῆς ἐπισκοπῆς σου
Psalm 137: 8 Wretched daughter of Babylon! blessed shall he be who shall reward thee as thou hast rewarded us. 9 Blessed shall he be who shall seize and dash thine infants against the rock. Compiled from the Translation by Sir Lancelot C. L. Brenton 1851
[Psa 137:8-9 LXX] (8) θυγάτηρ Βαβυλῶνος ἡ ταλαίπωρος μακάριος ὃς ἀνταποδώσει σοι τὸ ἀνταπόδομά σου ὃ ἀνταπέδωκας ἡμῗν (9) μακάριος ὃς κρατήσει καὶ ἐδαφιεῗ τὰ νήπιά σου πρὸς τὴν πέτραν
There is probably a general allusion here but let us not be too specific. The only words Luke 19:44 and Ps 137:9 have in common are "dashing babies".
However, the same pair of words also appear in 2 Kings 8:12; Isaiah 13:16-18; Hosea 10:14, 13:16, Nah 3:10, etc, and also in the Homer's Iliad, xxii. 63: "My bleeding infants dashed against the floor." (Thanks to Ellicott for this reference.)
This ancient barbarous behaviour was a sad but common practice of ancient warfare that has been replaced with more modern barbarism that is just as hideous.
There is an important difference between Luke 19:44 and Ps 137:9,
- Luke 19:44 is discussing the Roman army despoiling Jerusalem and its inhabitants
- Ps 137:9 is discussing and un-named army that would destroy Edom, the "daughter of Babylon" (v7 & 8) and kill its inhabitants.
Thus, dashing infants against the ground or rocks is general figure of speech denoting merciless warfare (Isa 13:18) by the marauding army against its unfortunate victims.
The idea that Luke 19:44 is an allusion to Psalm 137:9 must be rejected on two grounds. First, Psalm 137's curse is not a prophecy but a poetic expression of extreme bitterness. Second, the prophecy in Luke is uttered in sadness while the imprecation in Psalm 137 is made in a spirit of revenge.
Note the context of Luke, just prior to the verse mentioned in the OP:
As he approached Jerusalem and saw the city, he wept over it and said, “If you, even you, had only known on this day what would bring you peace—but now it is hidden from your eyes. The days will come upon you when your enemies will build an embankment against you and encircle you and hem you in on every side."
Compare this with the feeling of the psalmist, who opens his song with a bitter lament over Israel's suffering in exile and concludes:
Daughter Babylon, doomed to destruction, happy is the one who repays you according to what you have done to us.
Happy is the one who seizes your infants and dashes them against the rocks.
Historically, Babylon was not repaid for what it did to Judah. No infants were dashed on its walls in retribution. The Babylon Empire collapsed after the battle of Opsis and Cyrus the Great took Babylon unopposed. As mentioned, the psalmist is not uttering a divinely inspired prophecy here, he is expressing understandable human bitterness at those who had destroyed Jerusalem and forced him and his people into exile.
So no, Luke 22 does not allude to Psalm 137. Jesus loved Jerusalem. He did not curse it or wish its residents ill; nor did he bless those who would later murder Jewish infants. Rather, he wept over Jerusalem's fate because its people did not understand. The psalmist, on the other hand, hated Babylon and rejoiced in the prospect of retribution, which did not in fact come to pass.