$\newcommand\ep\varepsilon$Yes, these random variables (r.v.'s) converge to $0$ uniformly in $m\in[n-1]:=\{1,\dots,n-1\}$ almost surely (a.s.).
Indeed, the r.v. in question is a special case of $C_{n,m}+i S_{n,m}$, where
$$C_{n,m}:=\frac1n\,\sum_{k=0}^{n-1}X_k\cos\frac{2\pi km}n,\quad
S_{n,m}:=\frac1n\,\sum_{k=0}^{n-1}X_k\sin\frac{2\pi km}n,$$
where $X_1,X_2,\dots$ are i.i.d. r.v.'s with finite absolute fifth moment. Let also $Y_k:=X_k-EX_k$. Then
$$C_{n,m}=\frac1n\,\sum_{k=0}^{n-1}Y_k\cos\frac{2\pi km}n,$$
because $\sum_{k=0}^{n-1}\cos\frac{2\pi km}n=0$ for $m\in[n-1]$.
So,
$$EC_{n,m}^2=\frac{EY_1^2}{n^2}\,\sum_{k=0}^{n-1}\cos^2\frac{2\pi km}n=O(1/n);$$
the constants in $O(\cdot)$ here and in what follows depend only on $E|Y_1|^5$.
So, by Rosenthal's inequality (see e.g. this),
$$E|C_{n,m}|^5=O(1/n^{5/2}).$$
So, by the union and Markov inequalities, for all real $\ep>0$,
$$P(\max_{m\in[n-1]}|C_{n,m}|>\ep)
\le\sum_{m\in[n-1]} P(|C_{n,m}|>\ep)
\le\sum_{m\in[n-1]} \frac{E|C_{n,m}|^5}{\ep^5}
=O\Big(\frac{n^{-3/2}}{\ep^5}\Big).$$
So, by the Borel--Cantelli lemma, a.s. the events $\{\max_{m\in[n-1]}|C_{n,m}|>\ep\}$ occur only for finitely many $n$. That is, a.s.
$$\max_{m\in[n-1]}|C_{n,m}|\to0$$
(as $n\to\infty$).
Quite similarly, a.s.
$$\max_{m\in[n-1]}|S_{n,m}|\to0.$$
So, a.s.
$$\max_{m\in[n-1]}|C_{n,m}+i S_{n,m}|\to0.\quad\Box$$
Remark: Actually, it would be enough to assume that $E|X_1|^p<\infty$ for some real $p>4$.
