Английский Этимологический словарь - flush (v.)

- 1548, "A very puzzling word" [Weekley]. Original sense was "fly up suddenly," from c.1250 fliesen, perhaps related to flash via its variant flushe and thus connected to Du. vlacke. The sense of "spurt, rush out suddenly" is probably derived from this meaning, as is the sense of "go red in the face" (1630), through a shared notion of "sudden movement."