Quick format should be just fine in your case as the drive is new and, presumably, has no errors. The only time you really need to do a "full" format is when there's a bad or corrupted area on the disk/drive that you need to have overwritten and corrected.
A short explanation for what goes on in a quick format vs a full format is that there's a lookup table of sorts in a specific area of each disk/drive that lets the computer know where all the files are. A full format wipes that lookup table and all the rest of the files on the drive. A quick format simply wipes/changes the format of that lookup table without changing the actual data on the drive. Of course, the computer doesn't really care that there's still data on the drive after the quick format as the lookup table shows that it's just free space, and the computer will overwrite the old data as you go along.