Do people say "boy group" and "girl group" for Western groups?

We do, but the trend for Boy Groups and Girl Groups in the West is way down at the moment.

The last BIG Western Boy Group was probably One Direction and Fifth Harmony or Little Mix.

yep they do

I think I've heard the words "boy band" being used more than "boy group" for groups like 'NSYNC, Backstreet Boys, etc, when they were introduced in TV shows to do performances.

But, I cannot recall if "girl band" was used for girl groups like Destiny's Child, TLC, etc.