No, your existing installation cannot be integrated with QuickInstall (nor Fantastico). However, you can accomplish your goal with a little work.
Pre-procedure check list
Make sure your script/program is available in QuickInstall. You can check here: QuickInstall Overview
You will need to install the program you want via QuickInstall, and then migrate the data from your old installation to your new installation.
Let's say you want to do this for WordPress; here are the general steps:
- Login to your control panel
- Move your current (old) installation of WordPress to a new folder
- Enter QuickInstall
- Choose to install WordPress to the original folder
- Use the same settings as you did for your old WordPress
- Create a SQL backup of your old WordPress database
- Import that SQL backup to your new WordPress database
- Visit your blog and refresh to be sure every thing looks the same
Now you can manage and update WordPress via QuickInstall. This procedure works the same for any program you want to hooked up to QuickInstall.