Social media can act as a sentinel to inform governmental agencies of citizens' attitudes and to detect signs of distrust. A number of studies have used qualitative research methods to investigate the role of social media in building and maintaining trust relationships between governments and citizens. Identifying quantitative indicators of government trustworthiness on social media is a difficult and little-explored question. Although engagement metrics have been used to gather insights on governmental use of social media, they are usually not grounded on theoretical trust frameworks. In this study, we propose to characterise the governmental presence on social media through a set of indicators, computed directly and automatically from social media data, that can be mapped to dimensions of trust drawn from trust literature. An application of our framework on two public health authorities' Twitter handles is also presented.