It's grammatically OK, but unusual. As the other native-speaker members have said, it's more usual to say 'I've never surfed' or 'I've never been surfing/gone surfing'. You could say 'I've never done surfing' is if you are looking down a list of various activities including nouns which go with 'do', saying, for example 'I've never done yoga', 'I've never done ...' and then this would fit into the list. It would sound reasonably natural in that context. The other time you could use 'done' is if you are treating 'surfing' as a kind of uncountable noun, in sentences such as 'I've never done any surfing' or 'I've never done much surfing' or 'I've never done a lot of surfing'.

I've never surfed, or, I have never gone surfing.

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You could say "I've never surfed", but "I've never surf" is incorrect.

It's hard to say that it's grammatically wrong, but it doesn't sound right to my ear. I think the more natural way to say it is "I've never been [or "gone"] surfing".