The short answer is no.

The student is generally considered an employee but services performed by a student who meets certain conditions are specifically excepted from the definition of "employment" for purposes of FICA under §3121(b)(10).  SECA under §§1401 and 1402 does not apply to the student.

If you are interested, you may refer to §31.3121(b)(10)-2 for the specific conditions that apply to this FICA exemption regardless of whether the stipend is subject to income tax.

Colleges and universities are usually very in tune with these tax provisions as they regularly disburse various payments to students under different arrangements.