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Um, this feature is already there and documented. The plugin is enabled by default. 

This feature has actually existed since 2010. Not only Akeeba Backup was the first extension to implement control panel icons (even back in Joomla 1.5 when Joomla had no built-in support) but I am also the developer who contributed the Quick Icons feature to the control panel for Joomla 1.7.

For your use case, however, I would recommend using custom menu items on top of that icon plugin. By doing that you could actually have a “BACKUP NOW’ menu item at the top of the side menu which is a much more convenient place for your users since it's something they see in literally every page of the site. 

I am afraid this is not a feature we are going to implement.

Joomla's quick icons are not modules, they are plugins. While you can create multiple instances of a module you cannot do that for a plugin. A plugin would have to return the icon definitions for ALL backup profiles. As a result you'd have a long row with all Quick Icon backup profiles.

Having each Quick Icon backup profile able to display separately would require creating one new plugin extension per backup profile. First of all, this is problematic because we have to create installable, executable PHP code. We would also have a problem with how exactly these extensions are supposed to be updated when you update our software or removed when you either remove the backup profile or remove the package. They would also always appear as incompatible with newer Joomla versions because these are “orphan” extensions which do not belong to a package.

These are all points that you are blissfully unaware of as a user but we know as developers.

Creating a backup icon module would not make any sense either. You can already do that yourself using the Custom module type. The link you need to use is the same link you'd get in a custom backend menu item: index.php?option=com_akeebabackup&view=backup&autostart=1&porfileid=123&akeeba_hide_toolbar=1 where 123 is the backup profile ID. Since it's a custom HTML module you can do any kind of styling you want, not limited to how Joomla styles the Quick Icons. If you want to replicate Joomla's quick icon styling just right click on an icon and view source. It is something like this:

<nav class="quick-icons px-3 pb-3" aria-label="Quick Backup Links">
	<ul class="nav flex-wrap">
		<li class="quickicon quickicon-single">
			<a id="quick_backup"
					href="https://www.akeeba.com/administrator/index.php?option=com_akeebabackup&view=backup&autostart=1&porfileid=123&akeeba_hide_toolbar=1">
				<div class="quickicon-info">
					<div class="quickicon-icon">
						<div class="icon-akeebabackup" aria-hidden="true"></div>
					</div>
				</div>
				<div class="quickicon-name d-flex align-items-center">
					<span class="j-links-link">⚡️ QUICK BACKUP</span>
				</div>
			</a>
		</li>
		<li class="quickicon quickicon-single">
			<a id="full_backup"
					href="https://www.akeeba.com/administrator/index.php?option=com_akeebabackup&view=backup&autostart=1&porfileid=345&akeeba_hide_toolbar=1">
				<div class="quickicon-info">
					<div class="quickicon-icon">
						<div class="icon-akeebabackup" aria-hidden="true"></div>
					</div>
				</div>
				<div class="quickicon-name d-flex align-items-center">
					<span class="j-links-link">🐢 FULL BACKUP</span>
				</div>
			</a>
		</li>
	</ul>
</nav>

Finally, as I already said, you can create a custom backend menu which makes FAR MORE SENSE. I am not being difficult here, I am telling you what my observation is from real world users. The control panel icons move around and change colors, confusing the less than technically inclined among the users. Give them a menu item at the top of the side menu with a relevant Emoji as its first character and they can always find that link. For example, they will always be able to find the “⚡️ QUICK BACKUP” and “🐢 FULL BACKUP” menu items at the top of the sidebar menu.

There is no reason for us to implement a feature which is either going to cause massive problems or is a duplicate of existing core Joomla features.