Using the framework of Tutte embeddings, we begin an exploration of \emph{quantum graph drawing}, which uses quantum computers to visualize graphs. The main contributions of this paper include formulating a model for quantum graph drawing, describing how to create a graph-drawing quantum circuit from a given graph, and showing how a Tutte embedding can be calculated as a quantum state in this circuit that can then be sampled to extract the embedding. To evaluate the complexity of our quantum Tutte embedding circuits, we compare them to theoretical bounds established in the classical computing setting derived from a well-known classical algorithm for solving the types of linear systems that arise from Tutte embeddings. We also present empirical results obtained from experimental quantum simulations.