For eg, If Emotional intelligence and Social Intelligence are two dependent variables in a study, is it possible to test the association between emotion intelligence and social intelligence?
If your question is, can you correlate two variables, the answer is yes (of course!). Doing so would be a useful descriptive statistic for summarizing your data set, no matter what other analyses you applied.
As Safaa suggests, if your principal analysis concerns whether groups differ on these variables, it would likely make sense to analyze them simultaneously (e.g., in a multivariate comparison) rather than separately.
Yes, I agree with David and Safaa: you could correlate the two dependent variables to see if there is an association between them, assuming that whatever 'question' or 'intervention' that prompted the collection of the two DVs is identical.
As for terminology, if your hypothesis is that the two DVs will exhibit a relationship, then this seems appropriate.