Output field with double quotes in twig
Try with the value of the field and then concatenated with " like the second way:
I tested it with the following:

"name" : {{ content.field_text_test.0['#context']['value'] }} &lt;br&gt;
"name" : {{ '\"' ~ content.field_text_test.0['#context']['value'] ~ '\"' }} &lt;br&gt;
"name" : {{ '\"' }} {{ content.field_text_test.0['#context']['value'] ~ '\"' }} {{ '\"' }}


Here is the result: