I suppose you want to find \( Cov[\hat{\beta}_0, \hat{\beta}_1] \) which are the estimators in simple linear regression with the traditional set up. I think you only need to know that

\( Cov[aX + b, Y] = aCov[X, Y] \) and \( Cov[X, X] = Var[X] \) to calculate the covariance.