You can download the acutal responses as well as the grades. Under quiz admin -> Results, there are several options. Grades is one, Responses is another. (And I guess from what you say that you are not interested in Moodle's built in Statistics report.)
Of cousre, downloading stuff from the web can be scripted (using tools like wget or curl etc.) If you know any scripting language, that might be an easy option.
Alternatively, if you know any SQL, you might be able to pull all the data you want directly from the Moodle database. This SQL query might give you a starting point: https://docs.moodle.org/dev/Overview_of_the_Moodle_question_engine#Detailed_data_about_an_attempt