No, not with the standard moodle. I am not sure if there is a plugin to do what you wish.
You can, however, use the Advanced Search tool which then displays by date. You could grade (rank) those posts made by your first due date with a different score, for example.
You also cannot tell Moodle to set a due date as "one per day." And there are many other schemes that one can envision.
Now, don't get me wrong, Moodle's forum tool is by far much better than what other LMSs offer. Moodle still doesn't do every conceivable method that instructors think about. I have seen this request to have a separate first post due date before, such as "an initial post to the instructor's question" followed by a different date for "replies to other students." So your request is not weird or anything like that.