Probably not. Each year the ingredients in the flu vaccine are different, because each year different strains of the influenza virus are going around. They have to make up a new vaccine each year, to be sure it contains all the right strains that will provide immunity for the kinds of flu that are circulating at that time. Since the vaccine for the seasonal flu for this year in the Northern Hemisphere has just been manufactured and released very recently, it would not seem possible for it to be already expired. If you use a vaccine from a prior year (the more likely scenario if the expiration date on the bottle has already passed), then you will not be fully protected against the strains of virus that will be causing flu this year and may be taking something that could be harmful in addition to providing no protection.
You should talk to those at the source of the vaccine to find out why the vaccine would be expired.