You should file this on your resident state return. Additionally, if this is your only income in the state, you may check the state filing requirements, if your income is below a threshold amount, you may not need to file at all (every state has it's own rules). Using TurboTax, after you determine the amount that is taxable, you can check if you're required to file Federal, using the table below: https://ttlc.intuit.com/replies/3302274
A single person, under age 65, doesn't need to file if gross income received during the year is less than $10,350. However, if you're self-employed (1099-misc/contractor), then you need to file if your net income was $400 or more.
In determining your state of residence, if you maintain your residence at a family home, then time spent at the school is considered a temporary absence and doesn't require you to report your income in the school state. Instead, you'd report your fellowship income on your resident state return. If your go to school and live in that state, and that is your resident state; then, you'd file there.