No, they’re not from Star Trek.
It has nothing to do with the airline (except possibly sharing the name because of the similar shape).
In 1977 a contest was held among the members to create a fitting name for the new drum corps. “Spirit of Atlanta” was the winning name, and a logo was needed for this new organization. Again, members were asked to offer design suggestions, which would eventually become t-shirt designs.
The Delta, as it would come to be called, was nothing more than an abstract ‘A’ for Atlanta to be used on t-shirts, letterhead, and the like. Today they are seen hanging from the neck of the corps members in a stack, representing the years marched. In 1979 the first delta was made by cymbal player Nora Bierce (now Nora Whisenant) after cutting a section out of her cymbals which had cracked (as tends to happen to them after heavy usage and exposure to the heat). Members of the drum line then took the cracked cymbals and made deltas out of them using hand tools. They were smaller than those worn by current members (about the size of a finger nail). Unlike the current tradition, every member of the corps did not receive one.
For many years following, members who had deltas were members of the drum line or friends with the drum line. Quantities varied based on the number of cymbals cracked. Those who had a delta would only receive one (total, ever) instead of one per year.
Beginning in the summer of 1997, each member was given a delta made from cymbals used by the corps at the end of each summer. This would continue through the 2003 season.
Beginning in 2004 the switch was made to stainless steel, due to the significantly higher cost of the cymbals (and the desire of the cymbal companies to receive broken cymbals back, due to the high cost of raw materials for the manufacturers).
While the cymbal material gave the deltas a certain character, the switch to stainless steel allows the corps to have each delta imprinted with the year on it as well as a designation of ‘Member’, ‘Staff’, or ‘Volunteer’. It also provides the opportunity for alumni who marched in years where the delta tradition had not yet taken hold to order a set of their own.