My "boy" cousin filed a fraudulence case at the immigration against my "girl" cousin claiming that my "girl" cousin gave him her necklace that she pawned which he got and paid for at the pawnshop a long time ago. Is it possible for him to do that even if there was no written agrreement?

This question makes no sense to me, your kind of all over the place. What does immigration have to do with it? How could she have given him a necklace that she pawned, or vice-versa?

I don't see any link to immigration, unless the person accused of theft is now going to lose their green card or something.

Are you saying you have 2 cousins and one accused the other of theft? It doesn't matter that it was bought at and re-sold to a pawn shop... what matters is the actual crime... the theft.

Finally yes, if someone gives you a gift and steals it back you can call the cops and report it as a robbery.