More than likely, yes, supplemental nutrients will need to be added to the system. The reason being that fish food is the only input to the system. The fish food does contain all the correct nutrients the plants will need but not in the amounts that will be needed. The food is formulated for fish growth and not the secondary purpose of being used as plant nutrients. The nutrients that may need to be added are to compensate for the low levels supplied by the fish food.
The main deficiencies that will show up are in potassium, phosphorus, magnesium, calcium and iron.
Potassium is the most common deficiency in aquaponics. Symptoms being cupped or puckering of leaves. This is seen in the older leaves first and may also show up as marginal chlorosis. Leaves wilting even when the temperature is normal can be a symptom due to stunted root growth. Look at using potassium sulfate for supplemental corrections.
Phosphorus deficiencies are typically seen as a purple coloring of the leaves and veins. Brown spots on the leaves that resemble the leaf being smashed and spindly growth are also signs of a phosphorus deficiency. Look at treating with chlorine free super triple phosphate or rock phosphate. (both becoming hard to find)
Next up is Magnesium. Symptoms for this are interveinal chlorosis starting on older leaves and some white to light brown necrotic spots. Older leaves will yellow at the edges leaving the center of the leaf green. Look for magnesium sulfate as a treatment unless the system pH is low then use calcium carbonate, which has magnesium in it.
Calcium deficiencies will leave the plant stunted, with cupped leaves, and interveinal chlorosis, showing signs in the new growth. If the plant is a fruiting variety the fruit may show signs by the presence of blossom end rot. Also look for brown spots on young stems and leaves. Calcium chloride can be applied via foliar spray to compensate for the deficiency.
Iron deficiencies are usually the easiest to figure out. The symptoms show up on all young growth as interveinal chlorosis. The chlorosis can be so severe that the symptomatic areas can be almost white. Look for an iron chelate as a treatment. You will need to use the correct one depending on the pH of your system. Above 7.5 use FeEDDHA, if pH is below 7.5 use FeDTPA.