It appears that both .dgr (CAD design) and .dgn (drawing) files originate from MicroStation and you need this software if you hope to somehow convert dgr to dgn or vice versa. Unfortunately, we haven't found anything specific about this conversion.
Convert MicroStation CAD design to MicroStation drawing.
We found one software available in the database for this conversion.
Microsoft Windows software - convert dgr to dgn on Windows
A CAD software from Bentley used for 2D and 3D design and drawing
Convert from dgr file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.