Get Toolbar's navigation icon view reference

You can make use of content description of the view and then use findViewWithText() method to get view reference

 public static View getToolbarNavigationIcon(Toolbar toolbar){
        //check if contentDescription previously was set
        boolean hadContentDescription = !TextUtils.isEmpty(toolbar.getNavigationContentDescription());
        String contentDescription = hadContentDescription ? toolbar.getNavigationContentDescription() : "navigationIcon";
        toolbar.setNavigationContentDescription(contentDescription);
        ArrayList<View> potentialViews = new ArrayList<View>();
        //find the view based on it's content description, set programatically or with android:contentDescription
        toolbar.findViewsWithText(potentialViews,contentDescription, View.FIND_VIEWS_WITH_CONTENT_DESCRIPTION);
        //Nav icon is always instantiated at this point because calling setNavigationContentDescription ensures its existence 
        View navIcon = null;
        if(potentialViews.size() > 0){
            navIcon = potentialViews.get(0); //navigation icon is ImageButton
        }
         //Clear content description if not previously present
        if(!hadContentDescription)
            toolbar.setNavigationContentDescription(null);
        return navIcon;
     }

More

Kotlin extension property:

val Toolbar.navigationIconView: View?
        get() {
            //check if contentDescription previously was set
            val hadContentDescription = !TextUtils.isEmpty(navigationContentDescription)
            val contentDescription = if (hadContentDescription) navigationContentDescription else "navigationIcon"
            navigationContentDescription = contentDescription
            val potentialViews = arrayListOf<View>()
            //find the view based on it's content description, set programatically or with android:contentDescription
            findViewsWithText(potentialViews, contentDescription, View.FIND_VIEWS_WITH_CONTENT_DESCRIPTION)
            //Clear content description if not previously present
            if (!hadContentDescription) {
                navigationContentDescription = null
            }
            //Nav icon is always instantiated at this point because calling setNavigationContentDescription ensures its existence
            return potentialViews.firstOrNull()
        }

After looking into Toolbar's child views in debug mode, I saw that drawer icon can be found there, as an ImageButton. (Thanks Elltz)

I use a Toolbar with custom xml layout with 2 children (LinearLayout and ImageView), so my Toolbar had 4 children in the end, with these positions:

[0] LinearLayout(from custom xml)
[1] ImageView(from custom xml)
[2] ImageButton(drawer icon)
[3] ActionMenuView(menu icon)

Knowing this, I can now use:

View drawerIcon = toolbar.getChildAt(2);

to get a reference to drawer menu icon. In my case, the position is 2. This position should be equal to the number of child view's in your custom toolbar layout.

If someone finds better solution please let me know.

If you just want to get the Drawable representing the toolbar navigation icon, you can do this:

Drawable d = mToolbar.getNavigationIcon();

You can get a reference to the ImageButton used for the toolbar's navigation icon by having a method as this:

public ImageButton getToolbarNavigationButton() {
    int size = mToolbar.getChildCount();
    for (int i = 0; i < size; i++) {
        View child = mToolbar.getChildAt(i);
        if (child instanceof ImageButton) {
            ImageButton btn = (ImageButton) child;
            if (btn.getDrawable() == mToolbar.getNavigationIcon()) {
                return btn;
            }
        }
    }
    return null;
}