No it is not related. Sarah from LLL posted that it takes 5 cups of coffee to really affect the breast milk. If you are having one cup then it shouldn't. I was fooled into thinking that coffee was affecting my baby's sleep but now I'm sure it was other factors. Sarah said that the effects of caffeine for a newborn is 96 hours. If your baby is already three months then I think it drops significantly. By the time the baby is 6 months, the effects of caffeine in the system is something like 14 hours. But again, this is after having something like 5 cups of coffee.
I have a French press so I make half decaf/half regular because I use US sized mugs which are probably two cups of coffee. It seems to work for me because then I can also nap when my baby sleeps without being wired on caffeine.