That is not an expression that would be used in Canadian English. If you say it like that, I would interpret it as meaning that both you and your friend are inquiring about something, and it has been decided that your friend gets to do it first.
If you say, "You must inquire after your friend's health" that's a possibility.
I would say, "You must inquire about the state of your friend's health."
"You must inquire about the health of your friend."