Is it ok to give gluten free formula to my month old baby ? it won't increase her chances of being allergic to gluten, right ?

Sure : The major brand formulas do not contain gluten. They are cow milk products which do not contain wheat. Labeling as "gluten free" is perhaps a saavy marketing tactic since it is a huge consumer trend. Gluten allergy or celiac sprue can present in babies as you add wheat to their diet around a year of life. The general population has a 1 in 100 chance of having celiac or gluten allergy.