If the bag is fully loaded, yes, it will probably tip the Quest. It all depends on how much weight you put in the bag and how low you sling it to the back. The lower the bag, the less likely you'll tip. Here are a few suggestions for defeating the tip problem:
-pack less in the bag
-let your child know not to get out until you give the all-clear
-use the small handle on the back of the bag near the changing pad to hold the bag on the stroller. In other words, stick one stroller handle through the small handle on the bag and have the bag hang on the side of the stroller. It's less likely to tip there.