The Chicago Manual of Style, along with many other style guides and websites, says no:
6.68 Indirect questions
An indirect question never takes a question mark.
He wondered whether it was worth the risk.
But the example sentences at Oxford dictionary show that many people do place a question mark there. See Verb 1.1.
Examples:
‘I wonder whether you have thought more about it?’
‘I've been considering my last question, and I wondered whether the church in the photograph is in Venice?’
But note that are just as many examples that don't have a question mark.
I would be surprised if people who use a question mark in such indirect questions actually use a rising intonation when saying the sentence.