Don't be afraid to apologize if you did. The closest people - parents, mother will definitely forgive you. And they are not forced to beg for location as sometimes what partners do. Relatives need your first step, then they will gladly rush to establish a relationship.
If too shy to apologize in person, do it in the letter. Send mom a beautiful card in which you write how much you love her and how sorry that I offended you. Write down why you did it. Maybe you've been angry at my mother, and that feeling grew, accumulated, and eventually blurted out in a normal conversation. Don't hide your feelings. If it thinks you care about what is happening. Because indifference hurts the most.
Once the mother gets it and read the letter - call her. Come to visit her, taking something delicious for tea. Bring in a gift of something that she had long dreamed of - a new perfume, modern Hairdryer, crockery, etc. Give her the gift after the conciliation of tea, so mom didn't think you want to bribe her with a present. On the contrary, you have to say all the kind words during the conversation, but before leaving, to surprise. Then mom will only have positive impressions about your visit and she will forget the resentment.
Don't delay the apology. Of course, even if you don't ask me, mom will continue to communicate with you. But the former heat will not. And you, and she will always be a feeling of incompleteness. It hurt to be sincere with each other as before. It is best to apologize the next day after the conflict. This is enough time to cool down and gather my thoughts. The longer delay straight talk, the less necessary it seems. And in the end not happening at all, each time distancing from each other loved ones.