There are a variety of tests that could be used for ascertaining significance of a conjoint model or significant differences between levels within the same attribute.
Since you are using Discover, then we are using Bayesian (empirical Bayes) estimation of the utility scores. Practitioners often apply frequentist statistical testing procedures to the individual-level utility scores (which you can export from Discover to something like Excel or R), though this isn't formally proper per Bayesian standards.
So, if your research involves passing the review of Bayesians who will be keen on seeing that proper Bayesian tests were done, then this wouldn't pass muster. But again, for practitioner purposes the frequentist tests are often used.
Please see the following link (I'm assuming you're using CBC rather than MaxDiff) to an article describing different tests you could perform. If you go the frequentist route, I'd suggest the matched sample t-test.