The rise was led by construction, manufacturing and retail activity, ASB senior economist Jane Turner said.
Annual average growth was expected to rise to 2.8% from 2.5% when the latest figures are released on Thursday.
Ms Turner said the ASB forecast was stronger than the Reserve Bank’s 0.8% quarter-on-quarter forecast in the August monetary policy statement.
"However, we do not expect it to significantly alter the official cash rate outlook. We continue to expect a rate cut in November to 1.75% and growing risk of another next year."
Westpac economist Sarah Drought is not as confident as Ms Turner, forecasting economic growth to rise 1% in the June quarter.
She also expected positive news on the current account, with the annual deficit expected to narrow to 2.6% of GDP, bolstered by a surge in goods exports in the quarter and an upward revision to services exports.
Indicators for several months had been pointing to strengthening economic growth in the June quarter as gains broadened beyond the construction sector, she said.
Business confidence perked up, retail sales posted the strongest quarterly gain since 2006 and primary production rebounded.
Detail data released recently reinforced the view as the construction, manufacturing and wholesale trade surveys all reported solid gains.
"This led us to revise our forecast for the June quarter from 0.9% to 1%. While this will be an upside surprise to the Reserve Bank’s forecast of 0.8%, we don’t expect it to fundamentally change their view of the economy and the outlook for inflation."
New Zealand’s current account figures are due out on Wednesday and Ms Drought expects the deficit to narrow sharply to 2.6% of GDP, the smallest deficit since September 2014.
The starting point in the March quarter should also improve by about 0.5% due to an upward revision to education exports, worth more than $300 million in the year to March, with some offsetting revision from higher imports.
"The sharp narrowing in the annual deficit is quite remarkable given the terms of trade in the June quarter was down 3.9%."
Instead, the goods balance would be bolstered by a surge in goods export volumes in the quarter, up 10%.
While some exports, such as fruit, were experiencing a bumper run, much of the strength in June quarter volumes looked to be a matter of timing after two soft quarters, especially for meat and dairy exports, she said.
Exports would make a sizeable contribution to quarterly growth in the expenditure measure of GDP although it was expected to be moderated by a large run-down in stocks.
Strong service exports, led by tourism, continued to support the current account.
Gross domestic product
Gross domestic product (GDP) is the broadest quantitative measure of a nation’s total economic activity. More specifically, GDP represents the monetary value of all goods and services produced within a nation’s geographic borders over a specified period of time.
