You can ask your client to create an empty File Geodatabase using their Desktop 10.4 and provide it to you, if that is all you need.
I would still suggest you check the Client and geodatabase compatibility, and make sure the kind of data you are creating\using is accessible using older clients or not. There is a list of things mentioned in that link, which may not be supported and hence leading to your client getting the error.
Example from that link:
- Older ArcGIS clients can open, query, edit, and save data in newer release geodatabases, but they cannot open datasets that participate in newer functionality. You will encounter the following error messages when you try to access a newer type of dataset from an older ArcGIS client:
The version of the Geodatabase client is incompatible with the dataset and cannot open it.
Failed to add data, unsupported data type.