A show off can be called a vain person or if you want to use slang a 'smart arse' Primary school children call each other 'smarty pants' / 'teacher's pet' or in high school a 'suck' if they show off too much knowledge at school.

I would propose an another sarcastic sentense "Smart aleck again?" as well. Maybe someone finds more right English phrases to similar meaning as "showing off your intelligence"?.To someone who annoys other people by trying to appear to be cleverer and more knowledgeable then them...Like me.haha.I'm kidding.=)

If you omitted this important word as 'an intelligence' in your phase 'showing off your intelligence', then you'd probably change the whole meaning of stuff.'Showing off' does mean to pose and exibit oneself, when someone is trying to become the center of attention and etc. I guess that slightly differs with your intention to express yourself.