Changes in the values of the fundamental constants μ, the proton to electron mass ratio, and α, the fine structure constant due to rolling scalar fields, have been discussed both in the context of cosmology and in new physics such as supersymmetry models. This Letter examines the changes in these fundamental constants in a particular example of such fields, freezing and thawing slow roll quintessence. Constraints are placed on the product of a cosmological quantity, w, the equation of state parameter, and the square of the coupling constants for μ and α with the field, ζx (x=μ, α), using the existing observational limits on the values of Δx/x. Various examples of slow-rolling quintessence models are used to further quantify the constraints. Some of the examples appear to be rejected by the existing data which strongly suggest that conformation to the values of the fundamental constants in the early universe is a standard test that should be applied to any cosmological model or suggested new physics.