You can add start and end dates with the Custom Fields Power-Up for Trello: https://blog.trello.com/trello-custom-fields but those won't integrate as dates within your calendar. There's currently not a way to do that, only a due date.
You could intrgrate a gantt chart to track start and end dates: https://blog.trello.com/automate-gantt-chart-project-roadmap-elegantt-trello
I would also like to have this in the calendar view. Is that any plan to release this feature. It's tough to see what's currently being worked on when the card only shows up on it's due date. Ideally it would show the full lifecycle of the card during the time it's being processed. Would love to have the ability to set date ranges on the card dates!