You can choose any day for your custom NPC's birthday. There's nothing stopping you! However, many players prefer to only have one birthday per NPC. Below are some of the most popular NPCs and their birthdays if you wish to avoid conflicts.
It's also usually a good idea to avoid festivals for birthdays, as they can make some NPCs impossible to gift, but on the other hand it can be a fun way to add flavor to your mod. ("You skipped the festival just for my birthday?!")
Vanilla Stardew Valley will only show one birthday on the in-game calendar even if multiple NPCs have a birthday that day. You can use a mod like Show Birthdays to see if a day has more than one NPC birthday.