I figured I'd write up the elementary observations here, since no one else has: If $g^k = \mathrm{Id}$ in $PGL_n$, then $g^k = a \mathrm{Id}$ in $GL_n$ for some nonzero $a$. So the minimal polynomial of $g$ divides $x^k-a$. Let $p_1 p_2 \cdots p_r$ be the factorization of $x^k-a$ over $\mathbb{Q}$. We can use rational canonical form to write down a $\deg p_i \times \deg p_i$ matrix $g_i$ (over $\mathbb{Q}$) with characteristic polynomial $p_i$. Then, for any nonnnegative integers $a_i$ such that $\sum a_i \deg p_i = n$, we can take the block diagonal matrix whose entries are $a_i$ copies of $g_i$. Conversely, if $g^k = a$, we can break $g$ into blocks according to the irreducible factors of $x^k-a$.

So, what remains is to analyze the degrees of the $p_i$. Let $K$ be the splitting field of $x^k-a$ over $\mathbb{Q}$ and let $G$ be the Galois group. Write $\zeta$ for a primitive $k$-th root of $1$ and $\alpha$ for a chosen $k$-th root of $a$ inside $K$. Then every element of $G$ is of the form $\zeta^i \alpha \mapsto \zeta^{ui+v}$ for $u \in (\mathbb{Z}/k)^{\times}$ and $v \in \mathbb{Z}/k$. So $G$ is a subgroup of $(\mathbb{Z}/k)^{\times} \ltimes (\mathbb{Z}/k)$. Also, $G$ surjects onto $\mathrm{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})$, so, for every $u \in \mathbb{Z}/k^{\times}$, the group $G$ contains an element of the form $i \mapsto ui+v$. The problem is to describe the orbits of such a group on $\mathbb{Z}/k$.

For example, when $k=4$, then $G$ is a subgroup of $\{ \pm 1 \} \ltimes (\mathbb{Z}/4)$. If it isn't the whole group (in which case $x^4-a$ is irreducible), then it is either the group generated by $i \mapsto -i$ (in which case $x^4-a$ factors as $(\mbox{linear}) (\mbox{linear})(\mbox{quadratic})$, like $x^4-1$), or the group generated by $i \mapsto -i+1$ (in which case $x^4-a$ factors as $(\mbox{quadratic}) (\mbox{quadratic})$, like $x^4+4=(x^2+2x+2)(x^2-2x+2)$), or else $\{ \pm 1 \} \times 2 \mathbb{Z}/4$, in which case (in which case $x^4-a$ factors as $(\mbox{quadratic}) (\mbox{quadratic})$, like $x^4-4$).

At this point, it isn't clear what to do next, and the question is also a bit unfocused. Here are some (in my opinion) natural questions:

Is is true that $x^k-a$ always has a factor of degree $\geq \phi(k)$? UPDATE: No. $x^8-16 = (x^2-2)(x^2+2)(x^2-2x+2)(x^2+2x+2)$, and $\phi(8) = 4$.

For fixed $k$, what is the smallest $n$ for which $PGL_n(\mathbb{Q})$ has an element of order $k$? It is NOT $\phi(k)$: We can build elements of order $15$ in $GL_6$ as the direct sum of elements of orders $3$ and $5$ in $GL_2$ and $GL_4$, even though $\phi(15) = 8$.

I don't have an example of a case where $PGL_n(\mathbb{Q})$ has an element of order $k$ but $GL_n(\mathbb{Q})$ doesn't. I imagine such a thing exists, but it would be good to have an example.