The point graph of a generalized quadrangle GQ(s, t) is a strongly regular graph Γ = srg(ν, κ, λ, µ) whose parameters depend on s and t. By a detailed analysis of the adjacency matrix we compute the Terwilliger algebra of this kind of graphs (and denoted it by T ). We find that there are only two non-isomorphic Terwilliger algebras for all the generalized quadrangles. The two classes correspond to wether s2 = t or not. We decompose the algebra into direct sum of simple ideals. Considering the action T × ℂX → ℂX we find the decomposition into irreducible T-submodules of ℂX (where X is the set of vertices of the Γ).