We prove lower bound estimates for the first nonzero eigenvalue of the Laplacian on a compact quaternion-Kähler manifold. For the closed or Neumann case, the lower bounds depend on dimension, diameter, and lower bound of scalar curvature, and they are derived as the large time implication of the modulus of continuity estimates for solutions of the heat equation. For the Dirichlet case, we establish lower bounds that depend on dimension, inradius, lower bound of scalar curvature, and lower bound of the second fundamental form of the boundary, via a Laplace comparison theorem for the distance to the boundary function.