Are you copying and pasting the object from Illustrator to InDesign? The best way to import Illustrator art into InDesign is by using File>Place. That keeps the Illustrator art in Illustrator where it can be edited and then updated in InDesign as needed. It also assures that the Illustrator art will remain as it was intended. If you must copy and paste then check in Edit>Color Settings for both programs that all color settings are the same.