In fact, it is possible to do line density in QGIS using the standard menus sending parameters to the GDAL executables.
Using gdal_rasterize using the -burn -add switch would create a raster of densities of lines or points, with no search radius as kernel densities and similar. Just how many points or how much lines within a raster cell.
gdal_rasterize is found in the menu Raster -> Conversion -> Rasterize. In the last text field, you press the pencil button and alter the text to include "-burn 1 -add" similar to below. Naturally, you change the -tr (resolution) parameters and the input and output name according to your situation.
gdal_rasterize -burn 1 -add -tr 200.0 200.0 -l lines C:/lines.shp C:/out_lines.tif
Jukka Rahkonen gave me the hints on the gdal_dev mailing list. I've been looking for this for literally years.
Concerning the raster cell values for rasterized lines, I am unsure of what the values represent. Apparently it is not number of lines or lenghts of lines.