In this article, a novel idea of domination degree and index are defined in a fuzzy graph (FG) using weight of strong edges. The strong domination degree (SDD) of a vertex u is defined using the weight of minimal strong dominating set (MSDS) containing u. Methods to obtain an MSDS containing a particular vertex are discussed in the article. Idea of upper strong domination number, strong irredundance number, strong upper irredundance number, strong independent domination number, and strong independence number are explained and illustrated subsequently. Strong domination index (SDI) of an FG is defined using the SDD of each vertex. The concept is applied on various FGs like complete FG, complete bipartite and r-partite FG, fuzzy tree, fuzzy cycle, and fuzzy stars. Bounds involving the SDD and SDI are also obtained. Applications for SDD of a vertex is also provided.