There is some research, actually, that directly bears on that. A couple of political scientists at UBC ran an experiment where they presented the demographic quota policy related to the Canada research chairs. What they found was that for the group presented with the policy, it reduced trust in the research the university produces. It did that for both liberals and conservatives to almost exactly the same degree.
That is a good example of the fragility of trust and legitimacy and how they respond to different policies, so yes, for that particular policy, I think there is evidence that it erodes trust.