I think you are using Geographic CRS not Projected CRS. Usually this hillshade output is due to you are not using a projected DEM such as UTM projection with a meter unit. You need first to reproject your DEM from geographic to UTM (or any desired projection with meter unit) and use Bilinear or Cubic resampling.
You can see the difference between unprojected hillshade and projected hillshade in the image below :

To reproject the DEM data to UTM projection (as an example), go to Raster → Projections → Warp (Reproject)

- Select the Input File (DEM File).
- In the Output File: Give a new name to save the output
- Source SRS is the spatial reference system of the source DEM. In this example, it is EPSG:4326 (it means WGS 84)
- Check the Target SRS (the target projection): click on Select and choose UTM with correct zone or any desired projection with a meter unit
- Resample Method: Select Bilinear or Cubic. Here Cubic is selected.
- Click OK.
- Use the projected DEM as an input to create a hillshade