For any integer N>=1, we consider a class of self-injective special biserial algebras A_N given by quiver and relations over a field k. We study the Gerstenhaber structure of its Hochschild cohomology ring HH^*(A_N). This Hochschild cohomology ring is a finitely generated k-algebra, due to the results by Snashall and Taillefer. We employ their cohomology computations and Suarez-Alvarez's approach to compute all Gerstenhaber brackets of HH^*(A_N). Furthermore, we study the Lie algebra structure of the degree-1 cohomology HH^1(A_N) as embedded into a pull-back of quotients of subalgebras of Virasoro algebras and provide a decomposition of HH^n(A_N) as a module over HH^1(A_N).