I think it sounds like you are setting a very nice groundwork for a future relationship with your new GC. BAby is only 4 months and that is very young.
My MIL is absolutely wonderful, she wasn't very involved in the early days but she did come and snuggle DS and she was interested in everything about him including nappies, feeding, sleeping... all that stuff that parents of small babies suddenly become obsessed with.
My MIL also never offers advice (even when she really really wants to!) but will if asked. She told use lovely stories about when DH was a baby, what his temperament was like, how he fed, lots of lovely details that let us compare DS to DH.
She wasn't around every week but she text me(not DH) regularly asking how DS was and rang us both to chat about him. All of this showed me how much she loved him and made me so happy that he had such an engaged grandmother.
Now he is 5 and boy does he love his Nanna. He has sleepovers there occasionally and spends up to a week during the summer holidays with them. They will randomly pick him up from school and take him for dinner and bring him back at bedtime. They have a wonderful relationship and DS is confident in their love for him.
So, the point of my loooong post is that it is hard now, but you are setting the foundations for a long and happy relationship with your grandchild. At this age supporting your son and DIL and being really interested in your GC will be the best thing you can do. I hope you end up being as close to this grandchild as you are with your others.