In a "professional" way - you can not. Jane may be envious of you getting the lead role, she may be sincerely troubled by your perceived incompetence (sorry!) or other character flaws. It does not really matter, since the two motivations can flow one into the other quite easily.
This is most likely personal, and there are a few ways around this. I'd suggest getting into humor mode and confronting her with a sincere smile on your face and a direct question: "Heard you don't feel I'm fit for the job. Wonder what makes you think so." You are not looking for answers or an extended discussion, mind, what you want to achieve is put a bit of caution into Jane's attitude towards you, and end the confrontation at the time of your choosing.
Just an addendum: you cannot trust her, never tell her anything that may be used against you, but you can work with her by establishing an air of imperturbability and moral superiority. Never get emotional, never give her any sign that you are weak in any respect.
As a team lead, you have to pull the right string in human character. Optimally, this would be your subordinates' trust in your leadership. Failing this (as is the case with Jane, as it seems), it is the usual mix of working for the wage and sharing the goals of your organization. You have to watch out for repeated invectives in your direction; if Jane doesn't fall in line, you'd have to have another confrontation, and if that doesn't help, escalate (provided there are other candidates to fill the vacancy).