It had nothing to do with the refugees - in fact, one of the last things Abbott had done as Prime Minister, last week was announce that Australia would take 12,000 of them. And Turnbull's policies are likely the same.
The reason Abbott was replaced was because he was electorally unpopular - since mid 2014, well over a year, the coalition government has been behind in the polls, to the extent that Labor would have won any election held in that time. Abbott had been twice challenged before, and both times had held on, largely by convincing the party room that he could turn the electoral mathematics around. He couldn't. The party hopes that with Turnbull as leader, they will win the next election - and while Turnbull is no conservative himself, a lot of conservative members of Parliament know that they'll still have more influence as part of a Turnbull government than they would if Labor took office. The bottom line is that the country cannot afford a Labor government and that was becoming almost certain.
Why was Abbott unpopular? Mostly because of a massive media and social media smear campaign from the left wing press (which is most of the press) who ignored anything positive he did, exaggerated everything negative they could find, and blatantly made things up. One example of this - two weeks ago, the Australian Border Force (which was recently formed by a merger of part of the Department of Immigration with Australian Customs) issued a clumsily worded press release that said that they were going to be checking the visa status of people in Melbourne. This is a routine thing that Immigration has done many times before - basically they do things like go through a workplace of some description and get people to show they have a right to work in Australia (basically if they are not Australian citizens, that they have a work visa). In this case, they were going to be targeting taxi drivers. Totally routine - has happened many times before - but the media immediately suggested that this was a Gestapo "Papers, please" operation directed at mass checks. It was totally untrue. Even worse, the Sydney Morning Herald and the (Melbourne) Age newspapers claimed that Tony Abbott had personally ordered this when, in fact, he hadn't even been aware that it was happening (it was a routine operation, not something the Prime Minister is consulted about). They retracted that false accusation but most people don't read retractions. In essence, the left wing media didn't like the fact we had a conservative Prime Minister and set out to destroy him by fair means or foul. And it worked. Low information voters swallowed the lies and spread them further.
To these left wingers, Turnbull is the 'acceptable' part of the Liberal Party and they've wanted him in power since the election - but only because they can't have their first choice of a a Labor government.
Yes, this is exactly what happened. You are absolutely right.