When exporting models to OBJ using the Model Derivative API then you'll also get an MTL file with information about the materials being used in the original model. Many 3d viewers supporting OBJ will automatically look for the MTL file that the OBJ file is referencing and use the information in it when showing the model.
This functionality has been added quite a while back, but thought it would be worth pointing out anyway.
For test purposes I loaded the exported OBJ file into OBJViewer and got the following results when exporting an Inventor part file:
This works the same way for other file formats as well, including Fusion 360 models.
You can also use our sample project to get the MTL file when exporting to OBJ: