As filmor commented, you could redirect stdout to some file.
nohup ./script > script.out &
or even redirect both stdout and stderr to the same file
nohup ./script > script.out 2>&1 &
or to a different one
nohup ./script > script.out 2> script.err &
To discard an output, redirect it to /dev/null
. The data is basically destructed, so no memory is used. See null(4). If you want to discard both outputs use:
nohup ./script > /dev/null 2>&1 &
nohup(1) notices when both outputs are redirected, thus doesn't create the nohup.out
file (provided none of the two outputs is a terminal, see isatty(3)).
I would also recommend considering using batch(1) e.g. with a here document
batch << EOJ
./script > script.out 2>&1
EOJ
Read the advanced bash scripting guide.