The lines under each of my examples are the content of $MAKEFLAGS. It works when I pass the option in a different order and even with alternative spellings for -s like --quiet. It's always the "s" flag:

$ make -s --jobs=4
"s --jobserver-fds=3,4 -j"

$ make  --jobs=4 --silent
"s --jobserver-fds=3,4 -j"

$ make  --jobs=4 --quiet
"s --jobserver-fds=3,4 -j"

$ make  --jobs=4 --quiet -n
echo '"sn --jobserver-fds=3,4 -j"'

With just the -j flag $MAKEFLAGS starts with a space so the first word is empty:

$ make -j4
" --jobserver-fds=3,4 -j"

I don't know if other make implementations work similar but I think we require GNU Make, too.