I believe the specific question, can they become addicted to TV, is difficult or impossible to answer. A 3-month-old doesn't have control of their faculties, schedule, or location at any given time. A lot of what "addiction" means has to do with choices a person makes, e.g. they neglect other areas of their life in favor of whatever they are addicted to. These don't really apply to babies.
I don't think the situation you describe is in any way harmful for your child. You are holding them in your arms and watching television. Their eyes and attention will be naturally drawn to the movement, changing colors and sounds coming from the TV. However, because you are holding them they are still getting interaction and attention from an adult as well.
As the other answer mentioned there is a direct link between poor sleep and screen exposure, so you may want to avoid any screens being on for a few hours before they sleep in the evening.
I believe the main issues with screentime and development have to do with situations where the screen is used to soothe a child or hold their attention without another person present. The TV is no substitute for human interaction, and small children need as much human interaction as possible.
As long as you are not simply plopping your child in front of the TV, but are instead holding, loving and interacting with them, it's hard to believe the TV will cause any problem. And if you are ever worried they are relying too much on the TV for stimulation, or are too focused on the TV, simply turn it off or go elsewhere and interact with your little one face to face instead.