Don't think there are any "rules", but you won't be popular with your fellow first class passengers.

I was in Delta One recently on a flight from Amsterdam to Seattle, and an infant in the Delta One cabin was squawking ear-piercing screeches for much of the flight. Nobody regardless of service class should expect to endure repeated and ongoing ear-splitting shrieks of infants (most people are reasonably gracious and willing to tolerate some short term noise), but you sort of think that if you're riding up front, you stand a good chance of being spared.

If you do bring your little one, at least act like you're TRYING to calm baby down if he/ she starts making horrific noises.