there might be hacks for doing that but, basically, no, you cannot do it with scribus.
you need to create a text frame for each colored block of text.
in scribus the background colors for text are rather new and is not completely implemented.
i made a few tries and scribus seems to "paint" the font maximum ascent.
the question is: how can scribus do better?
if possible without needing a overly complex UI and with a straight forward implementation?
... adding the feature is scribus will anway a rather tedious work, since the work has been done for the display on screen and for each of the export foramts (pdf, ps/print, png, ...)