You are safe!
My wife and I came here with a 3 month old, he's now 7 months old. Everyone loves kids here and you'll find people smiling at you and very friendly. I'm not sure where you are planning on walking but we spend time on the Corniche, downtown and the malls as well as Al Khor and we've never had any hint of a problem.

Just be smart, there are a lot of places that are dangerous, from my experience, Qatar isn't one of them.