Hi! I looked up quarterly taxes before. I can't find the link, but I remembered reading that for fellowships, if they start in September, you don't pay the quarterly for then until January 15th. You need to have actually had the fellowship for a quarter of the year to pay taxes on it.
Also, there is only a penalty if you don't pay quarterly taxes. I don't fully understand why, at the end of the year, I can't just pay what I owe, but that isn't really important. Additionally, if you work another job (part time), you might not need to pay quarterly taxes if you make a certain amount (the taxes you paid from this part time job MAY clear what you would owe for the quarterly taxes). Thus, if you owed X amount for quarterly taxes, but you over pay taxes for your part time job (and thus would have gotten a tax return), you may just break even. But this requires a lot of math.