Can I say "do some recreation"?

Engaging in some recreational activities during the leisure time is good for us.

Immersing ourselves in recreational activities during the leisure time is good for us.

Getting some recreation during the leisure time is good for us. (nod to @James K)

Treating ourselves to some recreation during the leisure time is good for us.

Granting ourselves some recreation during the leisure time is good for us.

They may feel a bit weird still, because "during the leisure time" already hints at recreation being present. As in, leisure time is almost like a form of recreation.

“Doing some recreation” is technically okay, but “recreation” (or “recreational activities” if you prefer) sounds better as the subject.

“Do” is best used to clarify a contrast versus “don’t” and rarely needed alone.

You can use "do some recreation", but a couple of alternatives to consider:

Do some recreational activities
Get some recreation

The first is to emphasise that you mean "being active". The second follows the pattern of "get some rest", and seems to have slightly more use, eg

Camping is a great way to [...] get some recreation.

Also, no article before "leisure time", say "... during leisure time ..."