Is mentioning to my friends the situations that occurred between my cousin and me considered backbiting him, especially since the intention is to relieve myself and not to defame him?

Gheebah (backbiting) is mentioning your brother with what he dislikes. If the matters you mentioned displease this young man, then it is forbidden gheebah. However, if they do not displease him, then it is not gheebah. Nevertheless, one should avoid speaking about such matters, for "from the excellence of a person's Islam is his leaving alone that which does not concern him." The ideal solution for your love for this young man is marriage. You may propose marriage to him, provided there is no fear of fitnah (temptation) and no khalwah (private seclusion). It is preferable that your guardian propose that to him.