Qgis uses gdal to access .adf rasters, Arcinfo Binary Grids (note: .adf files are also used for Arcinfo vectors). Reading the reverse engineered Arc/Info Binary Grid Format description page one sees
vat.adf: This relates to the value attribute table. This is the table
corresponding integer raster values with a set of attributes. I
presume it is really just a pointer into info in a manner similar to
the pat.adf file in a vector coverage, but I haven't investigated yet.
So, no Qgis probably can't read Arcinfo raster attributes. That document was written a long time ago though, so perhaps things have changed.
I don't have access to a vat-enabled raster right now so can't test, but I surmise the structure outlined in the Raster dataset attribute tables Arcgis help is likely quite similar to the vat.adf structure, for those wishing to plug the hole in the format description and perhaps extend the gdal aigrid format driver.