What you may be looking was called "excelsior" and was made from shaved aspen, pine or other softwoods. It was also used to fill upholstered furniture and stuffed animals (toys).
I think there are still people making it, but I do not have any details of how it is made. It may be a by-prodcut from some other manufacturing process.
Shavings would come from a timber thicknesser or moulding machine. Try your local cabinet-maker or value-adding sawmill.
Why do you need wood shavings? Small melons like honeydews or rockmelons can be packed tightly in a standard single layer box without any filler, and for watermelons in bulk boxes straw is usual.