Quantum numbers required to specify the position of an electron
First of all, the question is badly formulated. It asks about "position" but clearly means "orbital", which are two different things. (An orbital is a wave function, which depends on the electrons position: $\psi(\vec r)$.) Although the difference may be difficult to explain at high school level, this choice of words is imprecise and often leads to wrong ideas about the basic concepts of quantum mechanics.

The question is obviously about the 4 quantum number ($n$, $l$, $m_l$ and $m_s$), thus including spin. Whether spin is really required to describe the spatial part of an orbital is arguable. But if we stick with high school level chemistry and consider what is typically called restricted orbitals, then both electrons ($\alpha$ and $\beta$) share the same spatial part of the orbital, and the spin quantum number is unrelated to the "position" of the electron. But the spin quantum number is still required to "distinguish" the two electrons, what seams to be what the question has in mind. (I used quotation marks, because the electrons are actually indistinguishable.)

If we go beyond high school level and into actual quantum chemistry, then it depends on what type of orbitals are considered. I already mentioned restricted orbitals. But one may also use unrestricted orbitals, where $\alpha$ and $\beta$ electrons do not share the same spatial part. Here it is obviously necessary to include the spin quantum number.

One final remark: There are actually 5 quantum numbers that specify an orbital. Analogous to $l$ and $m_l$ of spatial angular momentum, in spin angular momentum there is $s$ and $m_s$. However for electrons we always have $s=\frac{1}{2}$, therefore explicit specification of $s$ is not necessary.