What does a p-value of 0 imply in a Ljung Box statistic? What does a p-value of 0 in the Ljung-Box statistic imply?  
When I use the multivariate time series and fit a VARMA model, the Ljung-Box statistic (applied to residuals) gives values larger than 0.05 for all lags but the first two which are 1.
On the contrary, when I use difference series, I get 0's for all the lags but the first two which are 1. 
According to my understanding, the 0's imply that correlation exists in the fitted model. 
 A: The null hypothesis of the Ljung-Box test is that the autocorrelations (for the chosen lags) in the population from which the sample is taken are all zero. (See this thread for some more details on the test and the distribution of its statistic under the null.)
If your p-value is below your chosen significance level, you reject the null hypothesis in favour of the alternative that at least one of the autocorrelations is not zero in population. 
When applying the test on model residuals, you wish not to reject as otherwise the model assumptions are likely to be violated (we typically assume the errors are not autocorrelated).

What does a p-value of 0 imply in a Ljung Box statistic?

When applied on model residuals, it implies the model errors are autocorrelated, and thus you might not trust the model output (if the model is built under the assumption of non-autocorrelated errors, which it normally is).
Edit: as indicated by Michael Chernick, a p-value would not be exactly zero as that would only happen if the test statistic were infinitely large. But the p-value could be close to zero for a sufficiently large statistic, and then the software could round it to zero.
But read this thread which warns against using the Ljung-Box test on residuals from ARMA models.
