I guess that one could be correct. I just don't want anything to come between us.

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Hi, Sinem. It seems like neither of your sentences is correct. Instead, it should be "I just don't want anything to come between us ". "come between somebody" is a phrasal verb which means to make people argue and feel angry with each other, when they had been friends before or to prevent someone from giving enough attention to something. Have a look at some more examples: Nothing will ever come between us now. I didn’t want to come between a husband and wife. She never let anything come between her and her work. I hope that'll help.

It's also possible to say "... anything to come between us".

I just don't want anything to get (in) between us.