Yes mother can sign the petition and can receive the alimony at time of second motion. See your mother can withdraw her consent before second motion if her alimony amount is not settled, your father cannot get a divorce without her consent in mutual consent divorce.
SO your mother and father can make a mutual divorce agreement wherein they can specifically mention about payment of alimony and transfer of house, there wont be any issue the court seek consent at second motion also if she is not present or doesn't give her consent the divorce is not granted so don't worry and can go ahead and sign.Just keep in mind to make an proper mutual consent divorce agreement or MOU which specifically mention the amount and payment term.
Further make sure that your mother receive the amount before or at time of second motion