Discrete Fourier Series - Time Index
The integer $N$ is the number of samples of $x[n]$ in one period. There need not be any relation to absolute time whatsoever, simply because $x[n]$ are just discrete data that are not necessarily samples of a function of time.

However, if the data $x[n]$ are samples of some continuous-time signal, then the time difference between two adjacent samples is, by definition, the sampling interval $T_s$, which is the inverse of the sampling frequency $f_s=1/T_s$.

In that case, the time corresponding to one period of $x[n]$, i.e., the time difference between two samples $x[n]$ and $x[n+N]$, equals $N\cdot T_s=N/f_s$.