In practice, No. Theoretically, Yes.
No, because there are several factors that could cause a file to be fragmented, if you execute one or three copy dialogs at the same time shouldn’t matter.
Yes, because it could happen that one file from one dialog only write a file partially and the second dialog write the next file directly after it, causing the first file to be fragmented. However, given how many writes a typical computers executes every day, this could be ignored.
I guess the best solution (if you want to be 100% sure) is simply executing the copy as fast as possible to get the job done fast, and in the night run a defrag cycle.
In case you are running Windows 7 you don’t even need to this because it will run DEFRAG once a month anyway.