It is not legal for tourist or visitors to export stones out of Greenland (according to the Greenlandic law.
If you legally have bought jewelry with stone in it and you have the receipt you can bring them with you out of Greenland.
If the export is related to research or similar you can apply for an export license here:
The Mineral License and Safety Authority
Tel: (+299) 34 68 00
Fax: (+299) 32 43 02