Temporal graphs are introduced to model systems where the relationships among the entities of the system evolve over time. In this paper, we consider temporal graphs, where the edge set changes with time and all the changes are known a priori. The underlying graph of a temporal graph is a static graph consisting of all the vertices and edges that exist for at least one timestep in the temporal graph. The concept of 0–1 timed matching in temporal graphs was recently introduced as an extension of the matching problem in static graphs. A 0–1 timed matching of a temporal graph is a non-overlapping subset of the edge set of that temporal graph. The problem of finding the maximum 0–1 timed matching is proved to be NP-complete on multiple classes of temporal graphs. We study the fixed-parameter tractability of the maximum 0–1 timed matching problem. We prove that the problem remains to be NP-complete even when the underlying static graph of the temporal graph has a bounded treewidth. Furthermore, we establish that the problem is W[1]-hard when parameterized by the solution size. Next, we introduce the concept of edge overlap graph for a temporal graph. Finally, we present a fixed-parameter tractable (FPT) algorithm parameterized by the maximum vertex degree and the treewidth of the underlying graph of the temporal graph using the concept of edge overlap graph to address the problem.