Would covid tests show positive for immune individuals?

The Mantoux test is used in Ireland and perhaps elsewhere, reportedly to test immunity to tuberculosis, in order to determine whether a tb vaccine is warranted. An amount of the tuberculosis bacteria is injected into the skin to determine the reaction. A sensitive skin reaction indicates that a tb vaccine may be warrented. Ireland’s seemingly many subclinical cases of corona virus disease, hundreds, leads to the question, If an individual has developed immunity to a disease would a test for the disease show positive?