Otdfbt

Let's say I'm new to a school or a place and I'd like to know the name of my colleague and I want him or her to be my friend.

From a movie, I heard a sentence like that:

Can we know each other?

From google, I think I can say:

Can we get to know each other?

According to AmE, can I use those sentences so that people know what I mean? Is there a difference between them? Is there a better or more correct sentence?

Socially this is awkward. It is not common to "ask to be friends". If you don't know someone's name, you can't know that you want to be friends.

Instead, you introduce yourself, talk about things that interest you both, and if you get along you might arrange to meet again later. If you tell someone your name it is natural for them to tell you theirs (so you don't need to ask for it directly)

While you are introducing yourself, you can say something like "I'm new to this school, so I'm looking for friends". This is indirect and doesn't put any pressure on the other person. But any direct request is likely to be awkward. What if the other person wants to say "no"?

“Can we know each other?” is a weird-sounding question; don’t ask it. It sounds like it has the meaning “Is it possible we know each other?” or “Are we allowed to know each other?” - neither of which is what you mean to ask.

“Can we get to know each other?” is a normal question that someone could ask; the meaning is what you think it is. But I agree with James K that you should probably start off by asking someone’s name and talking about things you have in common. If you were to open with “Can we get to know each other?” it would probably come off as forward and/or aggressive- they might think you are interested in them in a romantic way and want to take them on a date. It would be best to get to know them first by introducing yourself and talking about school (or whatever the place is where you’ve met).

As the previous answerers noted, both phrases are grammatical, but awkward socially if said during an introduction.

That doesn't mean there isn't a place for them (or similar phrases).

"I hope we can spend some time together" is a good way to signal a casual friend you don't know well that you are interested in a romantic relationship, but with low pressure.

"I hope we can be friends" is a good phrase for clarifying intentions. For example, if you get the feeling someone is interested romantically but you just want normal friendship. Or if there was some conflict or disagreement you just settled and you want them to know that you are open to friendship.

But with someone you don't know, the best way to make friends is to start acting like a friend. Offering help with something works if you see the person needs something, but you can just start with small talk and then introduce yourself.