All surveys produce unreliable information (my main point in The Problem with Survey Research)–including surveys that ask about trust in government. Here’s a supporting comment by Michael Lipsky that appears in the November/December 2014 issue of Public Administration Review, p. 807: “it’s long past time to stop using polling data on `trust in government’ as measures of the public’s view of the public sector. These are entirely unreliable measures of the public’s assessment of government capacity because polling questions on trust in government, at best, conflate the performance of public agencies with the broad behavior of political actors. Respondents to such polls are reflecting as much on whether they approve of policy makers’ decisions as they are on the execution of public policy”.
I also discuss trust in government and surveys in my blog post, “Survey after Survey“.