How would I replace a single quote (') with a backslash then single quote (\') using sed?

just quote the replacement

$ echo \' | sed s/\'/"\\\'"/
$ \'

e.g

$ cat text1
this is a string, it has quotes, that's its quality
$ sed s/\'/"\\\'"/ text1 > text2
$ cat text2
this is a string, it has quotes, that\'s its quality

Try the following:

sed -e s/\'/\\\\\'/g input > output

To prove that this works:

echo "Hello 'World'" | sed -e s/\'/\\\\\'/g

The output should be:

Hello \'World\'