It can, but it doesn't typically. https://www.healthline.com/health-news/flu-allergies-coronavirus-different-symptoms SOB (shortness of breath, for those reading who don't know) isn't usually as severe with the flu as it is with covid19.
If you or someone you know is experiencing SOB, you should contact your doctor promptly. If the shortness of breath is making you or someone else really struggle to breathe, you should contact 911 or your local emergency number. Let them know that you may have covid19, but not being able to breathe is a medical emergency.
From what I've read, shortness of breath is more of a corona virus symptom than usual flu. Do you have any other symptoms?