Will I have to pay for my ex-wife’s attorney’s fees if she takes me back to court for violating an agreement?

Much of what your ex-wife can or cannot do will depend on your divorce decree. 
If your divorce decree has a 'morality clause' which says that you cannot have any girlfriends around your child, then she can file a motion to enforce, the judge will tell you 'don't do that,' and then life will go on. Some judges will award attorneys fees, but some won't.  It just depends on the judge you are in front of of.  Some believe that attorneys fees are just part of the ugly proces of divorce, where others will want the offending spouse to feel a little burn of the enforcement action since they can really do little else to them. 
If your divorce decree is silent and has no morality clause, then you ex-wife will have little hope at getting any type of relief based on the facts that you describe.  She cannot enforce what does not exist.  She would only have a chance if you did have a new girlfriend and the new girlfriend was a danger to your child (like has a felony conviction for assault or sexual abuse).  If your child simply doesn't like your new girlfriend, then the dislike is not sufficient to obtain an injunction.
Whether your ex-wife files for a modification to enforce an injunction or obtain an injunction, the burden of proof will still be her burden of proof.  Part of her proof can be the child.  Judge's in Texas have the authority to interview the child in chambers and in private.  While in chambers, the judge can ask the child questions regarding any concerns related to any girlfriends. If the child discloses that you have had a girlfriend that is bad to the child, then this could be sufficient evidence of a violation--assuming that your decree did have a morality clause.
If your child is emotionally disturbed, you need to do one of two things:  (1) get child into counseling so that you and the child can address this issue, or (2) make sure that you do not have this child around any new girlfriends.  The second measure isn't just to protect yourself from a modification suit, but also to protect your girlfriend.  If your ex-wife is coaching this child to say things and those things are not sufficient to get the desired result, she could coach your child to make other, more severe, allegations against your girlfriend.... and then your girlfriend will be forced to defend a criminal case while you defend the modification suit.
The advantage of getting your child into counseling now is to find out what is going on before the situation escalates.  If this is a case of parental alienation by your ex-wife, then the counselor can help reduce the effects of alienation and assist you with a defensive theory should your ex-wife file a suit.