All schools you rank for the Match or select on your Regular Decision Form will have your complete QuestBridge application and all associated materials (essays, recommendations, etc.). Each school has its own policies on reviewing the QuestBridge materials in conjunction with the other materials you submit to them directly (such as the Common Application). You should not automatically submit the Common App without first reviewing the school's policies.
You are free to include additional information in the Common App that you were not able to include in your QuestBridge application. You should also ensure that all important information from your QuestBridge application is included.