Your poor grandmother, she has always supported you, and as a grandmother myself I bet she absolutely dotes on you.
Your mother arranged to take you out (did she perhaps do this deliberately to stop you seeing your grandmother on her birthday, seeing as she doesn't like her?)
You lied to your nan and said you were just spending the day with your husband and child (who you could see any time).
You then went out with your mum, who then posted the pics on socials (again, did she do this knowing your grandmother would see it?).
Your grandmother was upset that you let her down on her birthday, (and probably that she didn't get to also see her great grandchild)
And then you blow up at your gran because you felt told off, and instead of taking on board this unusual criticism from her, you react with anger and decide to never see her again?
Advice: when you commit to seeing your grandmother, go and see your grandmother. When she is gone you will realise that almost nobody in the world loves you quite like a grandmother does.
Also work on how you react to criticism because reacting with anger won't help you in life. If someone who genuinely loves you and is almost always supportive tells you off, you need to self reflect, not fly off the handle.