The aim of the study was to assess the effect of long-term administration of natural prebiotics: Jerusalem artichoke (topinambur, TPB) and inulin (INU) as well as one of the most popular antidepressants, fluoxetine (FLU), on the proliferation of neural stem cells, learning and memory functions, and the composition of the intestinal microbiota in mice. Cognitive functions were assessed using the Morris Water Maze (MWM)Test. Cells were counted using a confocal microscope and ImageJ software. We performed 16S rRNA sequencing to assess changes in the gut microbiome of the mice. The obtained results showed that the 10-week supplementation with TPB (250 mg/kg) and INU (66 mg/kg) stimulates the growth of probiotic bacteria, does not affect the learning and memory process, and does not disturb the proliferation of neural stem cells in the tested animals. Based on this data, we can assume that both TPB and INU seem to be safe for the proper course of neurogenesis. However, 2-week administration of FLU confirmed an inhibitory impact on Lactobacillus growth and negatively affected behavioral function and neurogenesis in healthy animals. The above studies suggest that the natural prebiotics TPB and INU, as natural supplements, may have the potential to enrich the diversity of intestinal microbiota, which may be beneficial for the BGM axis, cognitive functions, and neurogenesis.