1. "Could she possibly be married?"
2. Could she be married, by any chance?"
To the best of my knowledge, there is no basic difference between these two questions. Let's reword them.
The first question can also be:
Could she, by any chance, be married? All I just did was to break possibly which is an adverb into an interrupter. You will observe that it's not any different from the second question.
However, if you restructure the second question into this:
Could she be married, possibly?
Adverbs are very flexible, they can be put anywhere in a sentence ans still won't alter the meaning. But sometimes, they change the meaning of the sentence like the one above.
Could she be married, possibly? Here, possibly seems odd, and at best, it's a double question.
Let's give the scenario you painted a little background.
The presence of Lola is very intoxicating, at least for Ben. He casts another look at her. What a beauty. He has to do something about it. "Hey Luke," Ben turns to his colleague, "Could she be possibly married?" He nods Luke towards Lola who is wearing an elegant purple gown. He asks again, when Luke seems confused. "Could Lola be married? Possibly?"
In my opinion, there is no difference between the two sentences.