Short answer: sort of, yes. The New Testament definitely shows influence from the Greek (particularly) and Roman rhetorical and artistic traditions, but they're also innovative in a lot of ways. This is akin to the argument made in Eric Auerbach's extremely influential book Mimesis: The Representation of Reality in Western Literature. He focuses specifically on Peter's denial of Christ, and contrasts it against the works of writers (Petronius, Homer) who had gone before. He writes that the scene displays

something which neither the poets nor the historians of antiquity ever set out to portray: the birth of a spiritual movement in the depths of the common people, from within the everyday occurrences of contemporary life, which thus assumes an importance it could never have assumed in antique literature. . . . A scene like Peter’s denial fits into no antique genre. It is too serious for comedy, too contemporary and everyday for tragedy, politically too insignificant for history.

It's worth reading the chapter (and the book) in its entirety. A neat little digital humanities project from the last few years was computationally comparing the gospels' style to their antique forebears, and finds clear statistical differences.