I would like to know if my two-dimensional data set to follow any of the following distributions such as the mixed multivariate normal model, multivariate t model, multivariate non-central t model. Are there well-known tests to know if I may apply them to my data set?
A good answer would require knowing some context. The answer given in comments above is not entirely correct. To simulate from one of this distributions, you would have to first estimate parameters, using your data. But the KS test is only correct when the null hypothesis distribution is entirely prespecified, so using it (uncorrected) after parameter estimation is incorrect.
One possibility would be to estimate all of the distributions in your (short!) list by maximum likelihood, and compare them by information criteria like the AIC.