Can an irrelevant variable be significant in a regression model? I'm reflecting about statistical significance and I think a variable can be irrelevant and significant at the same time. But, how do I explain that? And if that were the case, should I include that variable to my regression model?

The definition of significance is that there is a certain probablility (not more than $\alpha=0.05$) that the wrong conclusion was drawn. The result of a statistical test can be significant, i.e. it can state that whatever the respective test tests for holds with high probability.
For example, the test result can state that there is a high probability that the weight of individual apples follows a Gaussian distribution. Or that there is a high probability that the weight of individual apples does not follow a Gaussian distribution. (Note that one of those two cases is wrong, i.e. the test yielded a wrong result. The probability for a wrong but significant result is not more than $\alpha=0.05$.) Or the test result might state that there is a high probability that ice cream consumption and shark attack frequency are correlated. Or that there is a high probability that ice cream consumption does not cause shark attacks. Or that there is a high probability that shark attacks do not cause higher ice cream consumption. Depends on which statistical test is applied to what data.
However, I don't know what you mean by a variable being "significant". The result of a statistical test can be significant. One variable can play different roles in statistical tests.
