In the Options > Customize dialog, when you set the "Choose commands from" dropdown to All Commands, you can find the three entries Apply Heading 1, Apply Heading 2, and Apply Heading 3. (These styles also have default keyboard shortcuts, Ctrl+Alt+1, 2,
and 3.)
Other than those three, there is no built-in way to put individual styles on the Quick Access Toolbar. You could create a macro for each style you want, similar to this one (see
http://www.gmayor.com/installing_macro.htm if needed):
Sub ApplyHeading4()
Selection.Style = ActiveDocument.Styles("Heading 4")
End Sub
Then, in the Options > Customize dialog, when you set the "Choose commands from" dropdown to Macros, you can find your macro's name. After adding it to the list on the right side of the dialog, use the Modify button to clean up the display name and choose an
icon.
If your need is only to have the styles available regardless of which ribbon tab is showing, you can add the entire Quick Styles gallery to the Quick Access Toolbar. Simply right-click anywhere in the gallery and click Add Gallery to Quick Access Toolbar. However,
this won't reduce the number of clicks needed to get to a style -- it's just as easy to click the Home tab and then pick a style.