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Hi, can we use unit root test of residual to detect autocorrelation in a time series model? Are stationarity of the residual means there is no autocorrelation?

To determine if residuals are serially correlated, you must use tests designed to test for this specifically. The most famous of these tests is the Durbin-Watson, but you should probably use the LM tests proposed by White, Breusch-Pagan-Godfrey, Harvey, and so on.

Note that if you are testing for unit-roots in the residuals, what you are effectively testing is for cointegration between the variables used in the regression which generated the residuals in the first place.

If there is cointegration between variables, is it means there is no autocorrelation in that model?

No.