I'm using IRS Topic 421 and IRS Pub 505 as references. Note that state tax withholding and estimated payment rules may differ from federal.
If you had 1) no tax liability for 2022, 2) were a U.S. citizen or resident alien for the whole year, and 3) your 2022 tax year covered a 12-month period, you don't have to pay estimated taxes for 2023. However, you'll still owe taxes for 2023 in April 2024.
Fellowship income as payment for services is fully taxable, with some exceptions for services required for certain programs (see tax topic 421 linked above). There may be exceptions for certain non-resident alien visa holders due to specific tax treaties as well.
If you expect to owe at least $1,000 for 2023 AND you expect your withholdings to be less than the smaller of 90% of your 2023 tax liability or 100% (110% if AGI >$150k) of your 2022 tax liability, you are subject to estimated taxes.
The "standard" way of paying estimated taxes is to make four equal payments on or before April, June, September, and January 15th such that your total payments equal or exceed the lesser of 90% of 2023's tax liability or 100% (110% if AGI >$150k) of 2022's tax liability. You can pay by check with a 1040-ES payment voucher attached, online through your IRS portal, or over the phone.
You're also allowed to make a late estimated payment. You'll be subject to penalties and interest up to the point the required payment is satisfied.