The concerts for Drax Project and Mitch James in Christchurch, Dunedin, and Hamilton are still scheduled to go ahead in July.
Drax Project shared the news with their fans on social media: "Hey Wellington, gonna have to postpone this weekend's shows due to the Covid alert level change, working on the soonest possible alternative dates!!"
Fans expressed their disappointment in the comments, with many saying they were "gutted".
The change in alert levels comes after a Sydney man travelled to the region while infected with Covid-19, including fears it might be the Delta variant.
Under alert level 2, gatherings are restricted to 100 people meaning several events in the capital will not be able to go ahead in the coming days.
The move to level 2 will prohibit crowds at tonight's Wellington Sports Awards, the Māori All Blacks v Samoa rugby match at Sky Stadium on Saturday, as well as home matches for the Saints basketballers and Pulse netballers on Sunday, Newstalk ZB reports.