The term ‘birthday’ is generally not considered a proper noun. Proper nouns refer to specific names of people, places, or organizations, and they always start with a capital letter. For example, ‘John’, ‘Paris’, and ‘Amazon’ are proper nouns.
‘Birthday’, on the other hand, is a common noun because it describes a general occasion that applies to anyone. While you might capitalize ‘Birthday’ if it’s part of a title or a specific event (like ‘Birthday Party for Lisa’), in everyday usage, it remains a common noun.
Thus, unless it begins a sentence or is part of a title, it should be written in lowercase.