The incident occurred between 4pm and 5pm on Sunday, July 29 at the Harvester restaurant in Whichers Gate Road.
The victim, a 61-year-old woman, was at the restaurant with a friend when a man and woman at an adjacent table targeted them with homophobic abuse and a threat of violence.
PC Brent Vine from Waterlooville police station said:
“This incident was very traumatic for the victim as it involved comments about her sexuality and resulted in the victim and her friend leaving the restaurant before they had eaten.
“I would like to hear from anyone who can help me identify the man from the picture as he may be able to help with our enquiries.
“I’d also like to speak to anyone who was in the restaurant at the time of the incident who has any information.”
Anyone with information about the incident is asked to contact PC Vine at Waterlooville police station by phoning 101.
Mini-Com users can call the police on 18001 101.
Information can be given anonymously by calling the Crimestoppers charity on 0800 555 111.