Note: At some point, you may need to utilize QuickShortcutMaker or Activity Launcher app.
You're definitely going to need root access to show or remove anything from the Quick Settings drawer.
Gravitybox (gratis; requires Xposed Framework) can do it. It is available for Android Jelly Bean, Kitkat, and Lollipop.
GravityBox is a module [whose] primary goal is to provide the users of AOSP devices
with a tweak box to turn their vanilla AOSP ROM into custom ROM packed with additional features and tweaks
without need to flash anything. Most of the preferences can be changed on-the-fly without rebooting.
The ROM needs to be close to AOSP in order for this module to work (ideally).
(Click image to enlarge)
- Launch Gravitybox → Statusbar Tweaks → QuickSettings management
- Go to Statusbar QuickSettings tile → select Quick App → OK
Go to Tile specific settings → QuickApp tile settings → Main application
Now, there are many choices for you.
- If the note taking app directly launches into an editor then choose the app under Applications.
- If it has a shortcut which can launch its editor then select the app under Shortcuts.
If none of those two options are available to launch the app's editor directly but a widget can do it, then use Popup Widget 2 (paid) to create a widget that can be launched from anywhere. See my answer for the question How do I launch a widget from a shortcut on Home screen?
After that choose Popup Widget 2 as shortcut under that Shortcuts options of GravityBox.
Alternatively, Overlays - Float Everywhere can also help.
An an alternative of adding an icon into the Quick-Settings drawer, you can consider Overlays - Float Everywhere. It can create a global sidebar where you can add your widgets. The sidebar remains hidden unless you show it by swiping from the left edge towards right direction. See more info on its website.
(Click image to enlarge; source: official website)
- Enable the sidebar from Home → Settings → Sidebar → slide Sidebar → configure as per your needs
- Thereafter, add a widget into the app. See my answer here for basic know-how.
- Close the app, go to home screen, drag the sidebar and add the entry.
Here is the official guide, should you need any help.
If the device is not rooted, then your best option is to use an app which uses a persistent notification as Quick Settings. I don't remember specifically any app, but Play Store is filled with such apps, launcher apps mostly.
One more alternative, try LMT Launcher. It is a freeware app and needs root access for some features. Use its global PIE navigation feature to access your note taking app or its editor.
(Click image to enlarge)
LMT is a tool for Android (tested already on multiple devices but should run on nearly all) that launches a command when [performed] single touch or multitouch gestures on the screen ([in] any app/screen/etc.). You can connect basic and advanced commands to any gesture LMT supports. In addition to the gestures you can activate a system-wide PieControl for navigation...
See my answer for basic know-how for the question Page up and page down in web pages and elsewhere.