That depends on the type of plant you are growing, but in general you can plant about twice as densely in aquaponics as you can in a traditional soil-based garden. This is because your aquaponics plants are getting exactly what they need at the root zone (nutrients, water, oxygen) so the plant doesn’t need to send its roots out searching for these things. The result is extremely compact, healthy roots and less competition for nutrients and water below “ground”. Your limiting factor is really how much light is getting to the plant. Check the space recommendations on the seed packet to get a better idea of how much room your plant will need at the full grow-out size, and know that these spacings were typically made for soil gardens so you can likely plant them closer. Just keep in mind that airflow/circulation is an important aspect of plant health AND pest control, so give everyone enough room to breathe freely!
Beware of companies selling aquaponics systems that set unrealistic expectations of how much food their systems can grow! Even though aquaponics does generally experience fast growth rates, it is, in the end, still a natural system bounded by natures rules.