When a new month comes along, your Hearthstone rank gets reset. If you had a win streak going the day before, are not higher than rank 6 (can you even get to rank 5 after a reset?) and win your first game, will you get 2 stars?
According to this PSA, no, win streaks will not carry over between seasons.
I just realized a few days ago that it's not necessarily obvious whether or not win streaks carry over between ranked months. I worked up a winstreak last night and then logged off. Just played rn and won my first match. No bonus star. I'm not that surprised, but I wasn't sure what would happen.