Jane may dislike Emma or just distrust her attentions for her past neglect but I don’t think there is any proof dislike of Emma caused her to share information about the Dixons, Weymouth or Frank with Mrs. Elton that she refused to share with Emma. It could be Mrs. Elton has asked Jane about the Dixons, Weymouth and Frank but if so (unless I missed it) it was off stage and I think it probable Jane was just as reserved with Mrs. Elton as she was with Emma on these topics. Is there any evidence to say otherwise? I think Jane is particularly reserved on these topics and if Emma had gone on to other subjects instead of preserving in these then perhaps she would have found Jane less reserved in her answers. Jane seems rather open in this exchange with Emma, basically saying she is not much interested in strangers but seeing (meeting with?) Mr. Elton will probably spark an interest:

"You are silent, Miss Fairfax -- but I hope you mean to take an interest in this news. You, who have been hearing and seeing so much of late on these subjects, who must have been so deep in the business on Miss Campbell's account -- we shall not excuse your being indifferent about Mr. Elton and Miss Hawkins."

"When I have seen Mr. Elton," replied Jane, "I dare say I shall be interested -- but I believe it requires that with me. And as it is some months since Miss Campbell married, the impression may be a little worn off." (21)

I was not suggesting Jane knows of Emma’s suspicions of her and Mr. Dixon. I was only describing Jane’s reserve on the subjects of the Dixons, Weymouth and Frank as disgusting and suspicious the same as Emma:

She was, besides, which was the worst of all, so cold, so cautious! There was no getting at her real opinion. Wrapt up in a cloak of politeness, she seemed determined to hazard nothing. She was disgustingly, was suspiciously reserved. (20)