In the movie, we see that Winona Kirk is onboard a starship where George Kirk Sr. serves. However, there is no indication that Winona would be working for Starfleet, directly or indirectly. She doesn't wear a uniform, isn't addressed by rank or position (or otherwise, beyond "sweetheart"), and isn't associated with any of the ship's personnel other than the medics caring for her and the husband caring for her. For all we know, she was aboard solely as a civilian passenger - certainly a possibility in all eras, and something we saw happen in ENT and TOS well before TNG made it explicit that civilian spouses and other family members can live aboard starships for extended periods of time.
I'm not saying that you're theory is impossible, merely improbable in my eyes. Since she was literally either in labor or holding her five minute old newborn in a medical shuttle for the entire movie and apparently was only spoken to by her husband, there is no evidence either way about what she was doing on the ship before heading down to sickbay to give birth. In an era where civilians did not really hang out on starships, it seems simpler to assume that absence of evidence to the contrary that she was in starfleet in some capacity or at least worked for starfleet. If she was another character giving birth in the 24th Century there would really be no way of guessing her status from what we saw. There aren't that many options in the 23rd however.