I had a similar issue, but had BOTH values I wanted to convert to no-data AND existing no-data values in the raster.
I came to this page for help, but I think there is a caveat to the approaches here. The methods of @underdark and @Micha only seem to work where there are no existing no-data values in the raster.
To get round this, you need to convert both the no-data values, and the values you wish to convert to no-data, to a consistent number. It is then okay to use the second step of @underdark.
To do this, use the Processing Toolbox > Reclassify Grid Values (SAGA) to convert the values and the no-data values to a common number (e.g. -999), at the same time. Specifically, use method "range" and specify the range. Then in replace no-data values, choose this same value (e.g. -999). Untick replace other values. This works for a single value change or more complex changes too.
Then follow the method of @underdark, and save the raster using Raster>Conversion>Translate, and tick no-data, entering the value you used (e.g. -999).