Can I get your contact details? ("details" means "all contact information": name, phone, email, address, etc.) Can I get your phone number? Can I get your email address?

I'm not sure exactly what you want, but I think you are asking the following. In an oral conversation ,you might ask somebody: "Could I have your contact information" This would usually be in a business conversation and not an informal new friend situation. In a casual new friend, social type atmosphere you might say: "I would like to stay in touch, can I have your --- phone number, QQ, e-mail etc-- .It would be your choice on what type of contact information you want. When you only ask for contact info it can mean many types. But in a business situation that is correct, they would give you there e-mail, phone and physical address typically on a business card. Info in an abbreviation for information and in a business letter you would look more professional spelling the whole word, information.