InDesign does not support Stylizing Fonts. If you apply a style to a font in QuarkXPress and the actual font does not exist, when you convert the file using Q2ID, the font will be flagged as missing. You will need to manually select a substitute font in InDesign.
In the example below, the BrushScript font does not have Bold or Bold Italic Styled fonts, so they are flagged as missing when converted using Q2ID, whereas Times Roman had Bold and Bold Italic Styles applied to it in QuarkXPress, and these fonts are available and thus not flagged as missing.
Stylizing includes applying All Caps and Small Caps as well. If the font does not actually contain a true All Cap or Small Cap glyph set, then the fonts will be flagged as missing.