Does quick tabs can be used on responsive design?

That will depend on how you use the tabs, what you put in them, the length of the tab labels etc. The module doesn’t dynamically generate fixed widths and heights, so as long as you know your way about the css that goes with responsive design then you shouldn't have many problems with this module in that respect. (I am building responsive Drupal sites featuring this module at the moment too).

You might be interested in #1934290: Quick Tabs to Responsive Accordion

Even simpler :

Download & install the tinynav.js module .

Go to admin/config/user-interface/tinynav

Replace (or add) the selectors:

.quicktabs-wrapper ul.quicktabs-tabs

And here is the drop-down quicktab menu.

For an "accordion" menu, the mobile menu toggle could do the trick replacing the default ID's or classes.

Problem in handling active tab otherwise #3 worked...
Issue is not solved hence I switched to MAC Quick tab style that is responsive in a way.