Since your phone doesn't have a 'Move' option, you will have to do it manually. This is the basic steps to export contacts as a .vcf file, import them to Google (aka Gmail) contacts and then sync them to back to your phone. The exact steps may be somewhat different for your phone. Consult your user manual for details.
1. Open the Contacts app and tap Menu.
2. Select Import/export, then Export to storage.
3. A box will open with the location the .vcf file will be exported to. Write it down. Proceed with exporting contacts.
4. Transfer the .vcf file to a computer or cloud storage for upload to Google. Or, upload can be done directly from the phone.
5. Log into your Google account and open Gmail Contacts. If you are uploading directly from the phone, you will need the desktop version of the Contacts webpage. You cannot use the mobile version.
6. Click on More. In the drop-down list select Import.
7. In the next box select Browse. Navigate to the exported .vcf file and select it. Click on Import.
8. Once your contacts are imported to your Google account, sync contacts to the phone.
You are done. Once you confirm that all your contacts have been successfully transferred you can delete the contacts in device memory.
Android since v1.0. Linux user since 2001.