It is not possible to customize the DateTime picker or the Calendar components. Instead of an "alert dialog box", I would go ahead and use the Dialog component, which can be seen as a kind of "custom dialog". From the docs: "A Dialog is a full-screen view that is displayed as a transparent layer on top of the current screen context. Basically, it's a separate view within the current application."
cascades_dialog.html?f=dial">https://developer.blackberry.com/native/reference/cascades/bb_cascades_dialog.html?f=dial
There is a fullscreen calendar event picker available on the platform...
cascades_pickers_calendareventpicker.html">https://developer.blackberry.com/native/reference/cascades/bb_cascades_pickers_calendareventpicker.html
...but there is no component available for you to use in order to display a month view kind of date time picker as shown in your picture.
So if you need an appearance exactly like shown in the picture, you will simply need to implement your own date/calendar components.
Also... This layout is very wide. Don't forget that you need a layout that runs well on the square 720x720 display of the Q10 and Q5.