Can fermentation and reproduction occur at the same time?
Yes, fermentation (I assume that you mean lactic acid fermentation) and reproduction can occur at the same time. A cell will go through the process of cellular respiration in order to produce ATP energy from glucose and oxygen. It's a two step process. The first step is glycolysis, and the second step is respiration. Respiration only occurs when oxygen is available. In the absence of oxygen, fermentation will step in and be step two during the process of cellular respiration.
It is entirely plausible that reproduction is occurring in an oxygen free environment. It's also possible that a person could be working anaerobically during reproduction; therefore, fermentation would be occurring.
I am suspicious of the original question though. Perhaps you meant to ask if respiration and fermentation can occur at the same time. The answer to that is "no." If oxygen is available, respiration occurs. If oxygen is not available, fermentation occurs.