I have luckily found a reliable way to temporarily get over the bug in which you can toggle the QuickNav feature of VoiceOver running on OS X 10.11 El Capitan.
This is a very simple and effective workaround on my MacAir 2014 and I just wanted to instantly share it with you since I know how bad it must feel to either keep switching to another application and back to where you were or have to restart VoiceOver madly just to get QuickNav to work again.
I was so excited to update to the latest OS X with its stunning improvements in efficiency and loads other under-hood privilege. Then as time goes by, I found myself getting mad at all time with the tendency to blow hot and cold of QuickNav feature, especially as I'm forgetting myself in browsing the web in Safari and have to switch back and forth and back and forth between applications.
Then I accidentally found its workaround -- as easy as a snap:
Whenever you found yourselves unable to perform the left-and-right arrow simultaneously pressed to get ease such as in web browsing with our dear friend QuickNav, just take it easy and press control -- yeah, that's it, press and release the key control alone before try performing the dear QuickNav call-key combination once more time. Chances are you will get connected to our dear friend again.
Wish you best of ease with our newly discovered solution and happy web-browsing