All GRASS color tables formats (v.6.4) are in this link:
If you have GRASS 6.4 in Linux, they are in /usr/lib/grass64/etc/colors.
For example, if you used the elevation table, this is the format:
0% 0 191 191
20% 0 255 0
40% 255 255 0
60% 255 127 0
80% 191 127 63
100% 200 200 200
Copy the above information as elevation.text and use the gdaldem command for producing your colored raster. The sintaxis of gdaldem color-relief would be (assuming your exported uncolored raster as input_raster.tif):
gdaldem color-relief input_raster.tif elevation.txt output_raster.tif
I ran the command in GRASS console. This is the result in my system when I loaded the output_raster.tif in QGIS. It can be observed that is a rgb image (corroborated with Value Tool Plugin).
You can do the same (but it is more difficult because there is not ColorRampItem definitions for the renderer) by using the Python Console of QGIS; as is exposed here: QGis Save Raster as Rendered Image