Given a distribution \(D\) of pebbles on the vertices of a graph \(G\), a pebbling move on \(G\) consists of removing two pebbles from a vertex and placing one on an adjacent vertex (the other is discarded). The pebbling number of \(G\), denoted \(f(G)\), is the smallest integer \(k\) such that any distribution of \(k\) pebbles on \(G\) allows one pebble to be moved to any specified vertex via pebbling moves. In this paper, we calculate the \(t\)-pebbling number of the graph \(D_{n,C_{2m}}\). Furthermore, we verify the \(q\)-\(t\)-pebbling number to demonstrate that \(D_{n,C_{2m}}\) possesses the \(2t\)-pebbling property.