Is there any tool or software except ArcGIS which I can use for converting coverage (.adf) files to shapefiles?
OGR can read ArcInfo binary coverages. If it is a vector coverage, and you have both the coverage directory AND the info directory (see coverage format) then you can use ogr2ogr to convert to a shapefile.
Edit: This assumes you are actually trying to convert a vector coverage, not a grid coverage. To check, look in the coverage directory, if you have files like hdr.adf and w001001.adf then it is a grid. If it has files like pat.adf or arc.adf then it is a vector coverage. You can also use ogrinfo or gdalinfo, ogrinfo will return information if it is a vector coverage and fail if it is a grid and vice versa for gdalinfo. If you really want to convert a grid coverage to a shapefile, then you could use gdal_polygonize
An .adf file in a coverage is a raster file, not a vector, so you can't go directly to a shapefile without coverting to a vector. You could use GDAL_Polygonize (available in QGIS) to do the conversion. This will take the raster and covert it to a vector.
Having said that it's possible, it's often not really a good idea unless the data really wasn't fit to be a raster in the first place. What are you hoping to do?
A coverage is a set of files. You can't just pick a .adf file out of a coverage. A coverage consists of all the files in that folder plus files in the info directory. I believe only ESRI software reads coverages.
I assume this is related to your other question about climate data, which is stored in raster format. Shapefiles are a vector format. Please research the difference between raster and vector data.