This doesn't really have anything to do with it, especially if you don't introduce yourself first. Some people swear by introducing themselves to that villager and then ignoring them, and others say it's best to talk to them a lot... in my own experience, the ones who have pinged me most are villagers I talk to from time to time but am not super close to. It's just random.
The method I use is this: find a date that is a Monday of a week with no events coming up. After a villager asks you to move and you say no, save your game, and set your 3DS clock to that date. Load as your mayor, save, and quit. Go back to your 3DS calendar and change the year to one year later. This time, start the game as a new character. After the train sequence with Rover, check the map to see who's missing. That villager will be the one to ping you next. Quit without saving and set the calendar back one year (so that you're on the exact same date you last played as your mayor). Load the game as your mayor and go talk to the villager who would be missing in the future. If they don't ping you, talk to them 5-6 times, save, quit, reload, and talk to that villager again. Repeat that last step until they ping you or mention moving in the conversation.
If it's someone you want to keep, tell them to stay and immediately save and quit. Set the calendar year forward by one year again, and this time, start as your mayor (nobody will have moved because you denied someone's request when you last played). Save and quit. Set the calendar forward by yet another year and use a new character to check who's missing. Quit without saving, set the calendar back one year, and then use the mayor to talk to the villager who would be moving next (like I explained above).
Keep doing that until it's someone you want to move out.
When it's the villager you want to move out, tell them to go and save and quit. You can now set the calendar back to your original date. Since going back in time by any amount counts as one day forward, make sure to note how many days the villager has to move out and subtract one (so that if the villager said they'd be moving in 5 days, they'll now move in 4 days).
It sounds complicated, but it's a safe method if you don't mess up when you have to start as a new character and it's pretty quick. Let me know if you need help or if you have any questions.