As the City Council begins a second set of hearings on the mayor’s budget this week, the question around City Hall is: How long will the honeymoon last between Mayor Bloomberg and Speaker Christine Quinn?

Traditionally, the relationship between mayors and speakers has been adversarial, but Mr. Bloomberg and Ms. Quinn have been nothing if not friendly since Ms. Quinn took office in January. The next six weeks will be a real test of whether they will work with or against each other in shaping city priorities. Few expect their sunny political relationship to make it through budget negotiations unscathed.

In releasing his $52.7 billion executive budget early this month, Mr. Bloomberg included two key concessions to the council. On Ms. Quinn’s request, he agreed to begin disclosing more detailed cost breakdowns of department programs and he earmarked $12 million for new bulletproof vests for police officers.

Despite the concessions, Ms. Quinn and other council members pointed to the mayor’s failure to restore $338 million in funding as a glaring omission in his budget. Lawmakers historically have had to lobby the mayor for that money before he puts it back in at the 11th hour; the council had hoped this year would be different.

“To some extent, the City Council was expecting the mayor to be kinder,” a Democratic political consultant, Scott Levenson, said.

Some lawmakers viewed the exclusion of the $338 million in funding as the end of the honeymoon period, the chairman of the council’s Finance Committee, David Weprin, said.

The annual battle over that lot of money lies at the center of what council members refer to as the “budget dance.” Even as she has largely avoided criticism of Mr. Bloomberg, Ms. Quinn says she is determined to end that ritual, arguing that it contributes

to unnecessary haggling over a tiny fraction of the total budget and weakens the council’s role in the public budget process.

If the mayor can be relied on to put the money in the budget by the July 1 deadline, why make an issue of it at all? Calling the administration’s bluff by refusing to advocate for the money is not the answer, Ms. Quinn said.

“This isn’t poker,” she said in an interview last week. “We’re playing with people’s lives, with people’s tax dollars, and we need to elevate the process to one of greater reflection, greater thought, greater stewardship – not diminish it to games or brinksmanship.”

As her colleagues prepare to question administration officials during budget hearings that begin Wednesday, Ms. Quinn is making appearances around the city to stump for initiatives she proposed. The most expensive is a proposal to spend $135 million over three years to convert all of the city’s half-day pre-kindergarten programs to a full day.

A spokesman for the mayor, Jordan Barowitz, said Mr. Bloomberg believes the money to expand the city’s pre-K programs would have to come from the state.

Mr. Bloomberg and Ms. Quinn have characterized their relationship as productive, but both are quick to say that they expect disagreements to arise. By law, the mayor and the council must finalize a budget by the end of next month. Mr. Bloomberg has proposed a conservative budget that uses much of a $3.4 billion surplus to pay down long-term debt and create a $2 billion health fund for retirees.

Despite the differences, political analysts say there is little sign that acrimony will rise to levels seen in recent years, when Ms. Quinn’s predecessor, Gifford Miller, threatened to have the council pass its own budget. Unlike Mr. Miller, Ms. Quinn is not running for mayor, and neither is Mr. Bloomberg.

A professor of public policy at Baruch College, Douglas Muzzio, said this year’s budget differences did not amount to “divorce material,” but that some conflict between the speaker and the mayor is to be expected. “I think they can do this civilly,” he said.