When the bus driver in the Bronx called out her name, Christine C. Quinn did not hesitate. The City Council speaker — and, as of Sunday, official candidate for mayor — bounded up the vehicle’s steps and gave him a warm handshake as passengers, unbothered by the delay, began to applaud.

“She’s got my vote,” the driver, William Adams, said as he started to ease his Bx5 bus back into traffic. “And everybody that gets on my bus.”

It was a scene nearly unimaginable in this buttoned-up era of City Hall: Mayor Michael R. Bloomberg, after all, is not known for greeting New Yorkers with an enthusiastic public display of affection.