I doubt that it would be appropriate and would be considered a bit odd. Most places, anyway. If there are no advertised positions, there are likely no positions at all. If there are advertised positions they will come with stated application procedures.
The department head normally isn't responsible directly for hiring faculty. It is a faculty committee decision and in some places the laws of the land require advertisements and fair consideration of candidates using published criteria.
I think it would be a waste of time to request a job. You don't say where you are, but I think that most in most places it would be improper to hire without advertising.
However, you might ask if the department contemplates hiring in the near future and where to look for announcements. But I wouldn't go beyond that.