Both gods have legends involving extraordinary powers as children.
This link discusses the infant Krishna, as an incarnation of Vishnu, instructing his father Vasudeva on how to escape the wrath of the tyrant Kamsa, and goes on to discuss the many wondrous deeds of the child god.
As the wiki notes, Krishna was also mischievous and a cow herder.
Hermes is likewise mischievous and perhaps the most famous cow herder in Greek mythology:
"He was laid in swaddling-bands on the winnowing fan,2 but he slipped out and made his way to Pieria and stole the kine which Apollo was herding.3 And lest he should be detected by the tracks, he put shoes on their feet and brought them to Pylus, and hid the rest in a cave..."
SOURCE: Apollodorus 3.10.2
Krishna is associated with music, famous for his flute playing, and Hermes is credited with the invention of the lyre, and plays a flute:
"...before the cave he found a tortoise browsing. He cleaned it out, strung the shell with chords made from the kine he had sacrificed, and having thus produced a lyre he invented also a plectrum ... when Apollo heard the lyre, he gave the kine in exchange for it. And while Hermes pastured them, he again made himself a shepherd's pipe and piped on it.
These are the obvious parallels—are there more? What have scholars written about this subject?