Often, both versions are interchangeable, but there can be a difference in implication...
1: He's happy doing that
strongly implies he's already doing it
2: He's happy to do that
implies he will do it - willingly (he's agreed to do it, but hasn't done so yet).
So given the cited example text, if we assume the reference is to studying English at degree level, the default assumptions would be that He is happy to read English means he's agreeable to applying to go to university to study English, whereas He is happy reading English implies he's already there on the course (and he likes it).
If "reading English" simply means reading English texts for pleasure (novels, poetry, etc.), it's unlikely anyone would use the infinitive form to read, because the construction [be] happy to do [something] is generally restricted to contexts where [something] is a task, rather than a pastime.