Marrying her is one thing; applying for and being granted permanent residence is another. Because she was previously in the US on a J1, she might be subject to the two-year foreign residence requirement. I would strongly suggest that you consult with an experienced immigration attorney near you for a full review and advice as to how you may proceed.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. A consultation with an experienced attorney is always the best way to go.
You get one clean shot - consult an immigration attorney.
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Certainly can marry but the question is whether a green card is possible. An attorney needs to determine if there was a two year.requirement on the J visa and if not what action to take. Also your status as Us citizen or permanent resident needs to be determined, which will affect the green card.