Let H be a connected labeled graph. In this article, we characterize all the total dominating sets and the minimal total dominating sets of H-join graphs. Consequently, we compute the (multivariate) total domination polynomial and the (multivariate) minimal total domination polynomial of H-join graphs. We also compute the total domination number of H-join graphs. Finally, as an illustration, we calculate the total domination polynomial and the minimal total domination polynomial of the join of graphs, multipartite graphs, Kn-join graphs, Kn1,...,nm-join graphs, the corona product of graphs and the windmill graphs.