There is little doubt that the nut trees mentioned in Song of Songs 6:11 are the trees known as the English, or more precisely as Persian, (Juglans regia) walnuts that are widely grown in many parts of the world. The center of origin of this species is not known but it is widely grown in the Middle East where the seeds of the fruits ("nuts") are eaten. A delicacy in parts of Syria is candied green walnut where the entire fruit including the part that will become bony, is eaten.
There are two gardens in Song of Songs 6. The first is in verse 2 and is a garden of spices and lilies. Verse 11 is a garden of nuts, located in the valley and associated with grapes and pomegranates. The speaker is looking down upon the garden in the valley. Scripture speaks much of the grape (more than any other plant) and often of the pomegranate but what does the walnut tell us?
The walnut is a large, graceful tree. The dark green leaves, and strong thick trunk would tower over the vines and the shrubby pomegranates. In ancient times, the walnut was a symbol of fertility.
The following from Greek scholar Eleanor Irwin (personal communication) is helpful. Generally all nuts and fruit are fertility symbols. The ancients knew that they were the end product of flowering. A nut like walnut with a big seed and a shell to crack must have seemed appropriate as an "encouragement" to throw towards the bride and groom (like rice).
The Greek bride and groom shared a quince on their wedding night as a symbol of the fertility (and sweetness?) they hoped for in their marriage.
Varro, who wrote a work in Latin on the Latin language, says that juglans "walnut" is from Jovis "of Jove or Jupiter" and glans "nut" and the walnut is Jupiter's nut. Presumably this signifies size and value. The walnut is the kind of nut that Jupiter would eat (a nut fit for the king of the gods).
Glans is cognate with Greek balanos "nut" often acorn, a common nut in Greece. The Greeks called the walnut Zeus' nut - Dios balanos. Dios is the genitive of Zeus]. So presumably the Romans copied the Greek idea that the walnut was Zeus' or Jupiter's, Zeus being the Greek equivalent of Jupiter or Jove.
Both glans and balanos are used for "nut-shaped objects" such as glans penis but I don't think the first thing that would come to mind in connection with the walnut would be Jove's nuts! By the time it became a single word, the etymology would be less obvious. The fact that Varro says what the components of the word are suggests that many people didn't realize it. For example, most English speakers don't realize that the daisy is "day's eye."