Britain still have three representatives - Andy Murray, Dan Evans and Konta - in the Grand Slam, but Watson and Edmund squandered the chance to progress.
It was a disappointing day for Edmund as he was beaten by Pablo Carreno-Busta 6-2 6-4 6-2 in just one hour and 48 minutes, landing just 53% of his first serves.
Edmund created just six break points during the entire match, but failed to take any of them as his opponent eased his way into the third round in Melbourne.
Elsewhere, Watson failed to follow up her impressive first-round win over Samantha Stosur by limping to a 2-6 7-6 10-8 defeat to American qualifier Jennifer Brady in two hours and 43 minutes.
Konta will continue flying the flag for Britain, though, as she overcame Japan's Naomi Osaka 6-4 6-2 in an impressive display, which included hitting 89% of first serves.
The 25-year-old will face 2011 semi-finalist Caroline Wozniacki in the next round.