In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total alpha-domination number and derive a lower bound for the total domination number of the product fuzzy graph in terms of the total alpha-domination number of the component graphs. A lower bound for the domination number of the same has also been found.