It certainly seems that the Greek invasion of India led to the introduction of the first statues of the Buddha. The Greeks loved to make statues of their deities, but the early Buddhists did not. See this answer on Buddhism Stack Exchange: Did the Buddha discourage antropomorphic representations of himself?
Alexander himself certainly seemed interested in participating in all the local religions around which he was encountering:
Alexander was unusually open to foreign religious influences and over the course of his lifetime he embraced many non-Greek deities and practices, sacrificing to the Egyptian gods while in Egypt and rebuilding the temple of the god Bel in Babylon.
Ultimately, it seems that the two cultures recognized the existence of the other as separate religions. There was enough cultural self awareness in this period in this part of the world for people to identify themselves as members of a particular religion, and ultimately to convert to another, such as the case of Greek King Meander I, who converted to Buddhism in the second century BCE. There is a Buddhist text, the Milinda Panha, which is philosophical discussions between King Meander and a Buddhist monk Nagasena.
So, ultimately, because of the self identification as distinct religions / traditions, there appears to be little narrative crossover in this period, as much as there were cultural crossover recognized in things like currency (see the Meander article again), art, and morality, e.g.:
In order to propagate rules based on morality, equality and righteousness (Pāli dhamma; Skt. dharma), he employed several official languages and scripts derived from Aramaic, namely Brāhmī and Kharoṣṭhī, and also used Greek and Aramaic in public inscriptions carved on polished rock or stone. His pragmatism was likely to have been inspired by the Buddhists who propagated their teachings in the languages spoken in different regions. In what may have been the earliest of these inscriptions issued in 258 B.C.E. Aśoka declares himself to have been a lay Buddhist disciple for more than two and a half years, and by the eighth year of his reign he publicly expresses remorse for the massacre in Kaliṇga and denounces the taking of lives.