Hi yoyo, I’m sure this is possible and for instance the upcoming event list is fairly close to what you want here insofar as it only displays current/future events, not passed events.
So you could make this the default view by changing the setting found at Events > Settings > General > Default view for the Events.
Of course, that is formatted pretty much as a straightforward list so you would need to spend some time customizing and perhaps tabulating the data to get a calendar-grid-like appearance.
Quite possibly there are other ways to tackle this, too, but I think personally I might start with this approach since altering the month view could be tricky, due to the month query being effectively hard-wired to pull data starting with the 1st day of the month.