This has been a very strange thread to read. Yes you could have worded your concern a bit better OP, as it's definitely not her 'job' to pick up your child, but I sense your frustration in your opening post and am surprised that most people have chosen to lynch you for that turn of phrase, rather than respond to your genuine concerns.
It does sound unusual that school would choose to only address issues with her and not you or his Dad. I would say it's down to one of three things. The first could indeed be that she's making it up, as you fear from past experiences. The second could be that she's specifically asking for an update on his day. Even then though I would say that, if something isn't major enough to mention to a care giver by choice, ie teacher asking themselves if they can have a word, then it doesn't need mentioning even if a report on his day is sought by whoever collects.
The third option is that it could be a different teacher on your grandmother's day, and they tend to be bigger on regular feedback than the usual class teacher for some reason. This would make most sense to me, as each teacher will have some release time during the week for planning, preparation and assessment (PPA). If you only ever see the class teacher and she only ever sees the PPA cover on her one day, neither of you will be aware that you're talking about a different person when she says his teacher said this...
Basically, you need to find out what is going on. Rather than just pick him up and assuming all is ok if they say nothing, actually ask "is everything ok? My grandmother often says she's been told he's not had a good day when she collects once a week, but I'm surprised that nobody has addressed this with me." See what they say. If they seem surprised, you know she's either lying/exaggerating or it's actually a different member of staff communicating with her, which school will be able to address if you put in a request that this information isn't shared with your grandmother and is sorted with a phone call to you if needed.
It's not an unreasonable request that they only discuss behaviour issues with you. As a teacher, I would only address this with parents/legal guardians anyway. I work in an area with high levels of EAL so we often have people say things like "please talk to me if there's a problem as her grandmother won't understand" etc. I know you haven't said that English being a second language is the issue here, but it's a similar request which schools will be use to hearing. No need for an email (which will go to the office or headteacher and could seem like a complaint), just tell the teacher in person. It's not an unreasonable or unusual request that behaviour issues are only discussed with the parents.