You normally "complain about something". Sometimes you use "of", e.g. She is always complaining of/about the weather.
You can use complain without a preposition.
If you say "I don't complain living here" (not very natural), it sounds as though you are saying that you have no objections to/you don't mind living somewhere. However, it would be more natural to choose between:
1) I have no complaints about living here.
2) Living here, I can't complain.
Other variations are, of course possible.
Note: you are normally asked to give a context when you give a sentence. I don't think it's necessary in this case, but often it's useful so that we can tell you the most natural thing to say.
You've not really given any context, but have only made your meaning slightly clearer!
For example, your home is a long way from the nearest town and someone asks you if you are happy living where you are. You would probably say "I don't mind living here" or "I can't complain" (meaning there is no real reason to complain).
I would not be natural to use "don't" with "complain" in this situation. Here is a situation where you might use "don't complain":
You live next door to someone who plays loud music for most of the night, which stops you from sleeping properly. A friend asks you why haven't complained to this person, and you reply "I don't complain because I know he is a very aggressive person and I'm afraid to make him angry." Or "I don't complain about the noise because........."