I think she did.
IIRC there's a story about her wearing a jewel that Henry gave her, and Anne gets very angry and pulls it off her. She didn't have to wear it, did she? Especially not around Anne...!
Also, Jane was a religious conservative (after they were married she knelt before Henry to plead for the lives of the Pilgrims of Grace and he said 'shut up, remember what happened to my last wives') so she might have thought morally she would be doing the right thing in encouraging Henry to get rid of dangerous heretic Anne.