Hey,

A food inspector is a government official who examines the safety and quality of food products, as well as the ingredients used in their manufacturing, the sanitation of the equipment used, and the location where they are produced.

Generally, the eligibility criteria for Food Inspector marks that candidates need to have completed their bachelors degree in the field of medicine, food technology, pharmacy, agriculture , dairy technology or any other relevant degree are eligible to appear for the Food Inspector Exam. The general age criteria for a food inspector is between 18-25 Years.

So, basically after doing hotel management course, you cannot go for the Food Inspector exam.  Know more about the course over here: https://www.careers360.com/careers/food-inspector