If this is your first time in Qatar, it is better to know the situation here first before bringing your child.
As you said, your contract is not Family Status.
In this case, you need to find an accommodation for both of you and your child.

You can always apply a resident visa for your child once you are here in Qatar but with your own expense.

Also, one thing you can do, try to bargain for a family status contract if you can with your employer.

Living status here is quite high, medicines are expensive too, and not to mention the price of the Baby's products.

Weight things up first.

Hope this will help.