When activating missing fonts, if InDesign mistakenly considers the fonts to be OpenType or substitutes with an unwanted font, you will need to install the correct fonts and either perform a Find and Replace or redo the conversion.
Another thing to keep in mind when fonts are flagged as missing is it may have to do with the new auto-activation and document based activation available under the later macOS versions. You may encounter timing issues when converting files using Q2ID, as Q2ID is building a new document without user intervention at speeds far faster than most people can move and this can cause fonts to be flagged as missing.
Sometimes just waiting a few seconds for the fonts to auto activate will cause missing fonts to “magically” appear correct. This is a timing issue and depends on your computer and if you are using TypeKit fonts or other web based fonts.