Detrend data with no clear secular trend prior to Fourier analysis?
How you pre-process data is related to the purpose of the analysis and since you haven't mentioned why you are doing spectral analysis, answering your question can't go beyond making some general remarks.

Detrending is a preprocessing operation. You want to compensate or remove some bias from your data. As an example, many kinds of data are collected in the presence of long term phenomena that is extraneous to what your are interested in finding. Many kinds of data have seasonal, diurnal, weekly, monthly, tidal, .... ect cycles associated with them. Unemployment has a seasonal dependence to it as an example.

I once was tasked with looking for jumps in syndrome data that is reported to health departments and there was strong increases on Mondays and after holidays in reports but that had to do with many doctor's offices being closed on weekends.  This was an artifact of how the data is collected and not how epidemics evolve.

Matlab's dtrend function removes a linear fit from data.  If the collection interval is small compared to some much longer trend in the data, it reduces the bias introduced by taking a section that is less than the period of the longer term trend, particularly for data like your which is strictly positive.  One may conceptualize dtrending as reducing low frequency spectral leakage that would otherwise mask features in your spectra.

Again, without knowing what you are looking for in your data, no specific answer can be recomended for your situation.