I think Dumbledore did believe at the time of Harry's parents' deaths that Black was guilty. So he didn't stand up for Black. But during the course of events in Harry Potter and the Prisoner of Azkaban, could Dumbledore have independently figured out that Black was innocent and this was merely confirmed by Harry. This would make sense Dumbledore, being a very intelligent man, wouldn't normally believe something without strong proof. Is there any evidence in the books for the same?
At first, yes!
Dumbledore actually testified that Sirius Black was their Secret-Keeper (via the Fidelius Charm). Therefore, he must have initially believed in Sirius's guilt.
I myself gave evidence to the Ministry that Sirius had been the Potters' Secret-Keeper.
Eventually, though, he comes around:
“But you believe us.”
“Yes, I do,” said Dumbledore quietly. “But I have no power to make other men see the truth, or to overrule the Minister of Magic...”