No, they are completely different characters.
As discussed in this Sci-Fi & Fantasy question and this Movies & TV question, Scarlet Witch and Quicksilver are rare exceptions to the general licensing deal that Marvel made with Fox. Part of that deal is that Fox has the exclusive rights to make movies about any of the "mutants" in the Marvel Universe: the X-Men, the Brotherhood of Evil Mutants, and all the related characters.
However, Scarlet Witch and Quicksilver are odd, in that they are also key members of The Avengers, which Marvel retained the rights to make movies about.
So, those two characters are legally allowed to be in both Marvel and Fox movies, as long as:
- Fox never uses the term Avengers or mentions any other Avengers characters
- Marvel never uses the terms mutant or X-Men, and in particular, never mentions Magneto.
So, the two movie universes have entirely different characters, with entirely different back stories, that just happen to be based on the same comic-book character.