Inter-rater reliability In my research, I use a questionnaire so that students mention some classmates according to a group of characteristics. Each item is a different characteristic, and for each one they must name the classmate they consider to have it. In this case can I use Krippendorff's alpha to assess the reliability of the students' judgments?

Krippendorff's alpha is a measure of (chance corrected) agreement. Reliability in classical test theory is the relationship between the true score and the measured score.  However, measures such as Kripp alpha are often referred to as reliability (but maybe I'm just old).
However, you have a weird issue, that people (presumably) cannot nominate themselves, so 100% agreement isn't possible.
