Can I date you on your birthday?
will you appoint me on your birthday?
Can you take you out on your birthday?
一般这样说比较符合英语习惯吧
Would you go out with me on your birthday ?
Could two of us have a date on your birthday ?
I'd like to date you(take you out) on your birthday, can I ?
个人认为语气很重要，在这种情形下需要委婉一些才好
请问您是男孩约女孩，还是女孩约男孩？抑或者是女孩约女孩？
大胆直接的说法：Can I date you on your birthday?
不过用date似乎有点太直接。
委婉点较好。如：Can I take you out on your birthday?
要含蓄的有：Can I fix an appointment with you on your birthday?
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