You can't configure diff-highlight to do that. It 'will find the common prefix and suffix of two lines, and consider everything in the middle to be "different".' [1]

To highlight only the quotes as changing, you need something else. On (Neo)Vim, for example, vim-gitgutter does this [2] (I'm the author).

[1] https://github.com/git/git/tree/master/contrib/diff-highligh... [2] https://github.com/airblade/vim-gitgutter/blob/master/autolo...