You can buy them after you get a certain amount of bagdes. You can buy them after you third or fourth badge. Quick Balls are also able to be found or given. If you want get free Quick Balls then go to Lumiose City.
"South Boulevard
Start at the far west end. You'll see an open door in the buildings. The people in here have advice... and nothing else. Sigh.
Down the street and on the left side is a cafe for photography enthusiasts. Photo spots should now be open to you.
Across from the cafe is another building. A girl in the lobby will give you a Quick Claw, while the computer workers on the second floor will hand over three Timer Balls and three Quick Balls."
I found it on a walkthrough.
Link: http://mattwritesstuff.hubpages.com/hub/Pokemon-X-and-Y-Walkthrough-Part-Six-Lumiose-City
