Do apprenticeships count as a student?

Do you mean a fellowship?

A fellowship is any amount paid or allowed to, or for the benefit of, an individual to aid such individual in the pursuit of study or research.

A scholarship or fellowship grant is tax free (excludable from gross income) only if you are a candidate for a degree at an eligible educational institution. Any amount of scholarship or fellowship that is greater than your education expenses is taxable.

Q. Re: Do apprenticeships count as a student?

A.  Simple answer: no.

But it depends on why you are asking.  

The expenses may be eligible for a tuition credit. Qualified educational expenses include "The expenses for fees, books, supplies, and equipment required for the designated beneficiary’s participation in an apprenticeship program registered and certified with the Secretary of Labor under section 1 of the National Apprenticeship Act. 6".  Reference: IRS Publication 970

For purposes of the dependency test:

"A full-time student is a student who is enrolled for the number of hours or courses that the school considers to be full-time attendance.

To qualify as a student, the person must be, during some part of each of any five calendar months of the year:

1. A full-time student at a school that has a regular teaching staff, course of study, and a regularly enrolled student body at the school, or
2. A student taking a full-time, on-farm training course given by a school described in (1), or by a state, county, or local government agency.

The five calendar months do not have to be consecutive." Reference: IRS Publication 17