No, contributions to his insurance are not taxable.
See this: https://www.irs.gov/taxtopics/tc421.html and https://www.irs.gov/publications/p505/index.html for information about estimated tax filing. It’s may not be just a once-a-year filing depending on his circumstances. Most fellowship recipients have to pay quarterly because they’ll owe more than $1k in taxes for the year.