You got four basic considerations, I think.
-- Who is she fleeing from?
If it's somebody powerful, and powerfully annoyed, the men and women in charge of the welfare of the convent aren't going to want to get mixed up in her quarrel. There is no profit to them and the possiblity of hazard, discomfort,expense, inconvenience and danger. It would be irresponsible of them to give refuge to somebody who had annoyed a powerful man if there were some way to avoid it.
Alternately -- if she has some potential protector, they're not going to want to irritate him by refusing refuge.
You see why Mother Superior is not going to be just delighted when this bundle of difficulty wanders in out of the night.
-- Who does she 'belong' to?
One great difference between here-and-now and there-and-then is that just about everybody in your there-and-then belonged to somebody who had the right to tell them what to do and to profit by their labor and who was responsible for their actions.
The very first task of the Mother Abbess will be to determine where this refugee fits into the great scheme of things. Is this a serf who must be returned to her master, a wife to her husband, an heiress to her guardian, an errant daughter to her father, an apprentice to her master, a bondmaid to her mistress, a nun to her own convent ...?
There were exceptions -- independent widows. Vagabonds. But just about everyone, especially every woman, was the property of someone.
The convent can no more 'keep' her than they could keep a strayed horse. Honesty, in that period, required that every effort be made to restore her to those who had authority over her.
-- You couldn't just join a convent.
Joining a convent involved, among other things, a transfer of authority from whoever the woman belong to, to the convent. One couldn't just walk in and ask to join. You had to be 'given' by your father or feudal lord.
The convent, accepting you, had to be sure you were free to take the vows required -- that you were not married or betrothed or otherwise committed elsewhere.
Those taking vows generally brought a dowery with them. This helped support the convent in which they lived. Even lay sisters would bring a small amount.
-- Convents were just hotbeds of benevolence.
All that said, all these considerations raised -- Convents were filled with kindly, humane women, independent, powerful, well-intentioned souls who might very plausibly intervene to save some suffering scrap of humanity.
Be aware of all the reasons the nuns couldn't help your girl,
but then, you can write it that they did
because there was almost certainly a lot of that going on too.