Which version of Android is on your LGE Optimus Q? According to the list of Adobe certified Android devices the LGE Optimus Q Flash Player 10.1 is supported on the Android 2.2 versions of your handset. I do not have access to and LGE Optimus Q device so you may need to tailor the following instruction to your specific device, but to check the Android version on your phone go to Settings > About Phone and the Android version should be listed there. If your phone is running Android version 2.2 you should be able find it by searching for Adobe Flash Player in the Google Play application store. If you are running a version of Android greater than 2.2 the Flash Player is likely not supported.
You can find a list of Adobe certified Android phones here: http://www.adobe.com/devnet-apps/flashruntimes/certified-devices.html