There are a couple of ways to change an object's opacity or fill alpha quickly using only the mouse, and not going via the Fill & Stroke dialog.
1.) right-click on the opacity indicator in the bottom left corner to be presented with a couple of preset choices for opacity, choose one of them
2.) use the scroll wheel of your mouse over said opacity indicator <- fastest option
3.) use a color gesture to change a fill's or stroke's alpha value by doing the following:
- hold the Alt key
- click on either the fill or the stroke color indicator field
- hold the mouse button down, and move the mouse on the canvas
- now move it relative to a diagonal line that goes from the bottom left corner of the canvas to the top right corner. The farther away from the bottom left corner you go, the finer will be the control you have over changes in alpha (try Shift for saturation, Ctrl for lightness, or no modifier key for hue).
If you need a keyboard shortcut, you'd need to write an extension that changes the alpha value of the selected object currently, and then assign a shortcut to the extension.
Or you could switch focus on the dialog (Ctrl+Shift+F), and use tab keys to navigate it, arrow keys to change values, and Escape to leave it again, if you do not want to use the mouse at all.
There's no option to assign a shortcut directly to this functionality in Inkscape currently. A list of all available actions that can have a shortcut can be found at Edit > Preferences > Interface > Keyboard Shortcuts.