1860 Manual of Algebra describes a method which is now taught in upper secondary schools worldwide:
To rationalize the denominators of fractions which consist of binomial quadratic surds, use the following RULE: Multiply the numerator and denominator by a binomial surd, conjugate in form to that which appears in the denominator.
I wasn't able to find an attestation of the term conjugate in this context earlier (there are a lot of conjugate diameters noising the search results though, and I could miss something), but plenty of people seem to be using multiplication by conjugate ante litteram, cf. these 1813 and even 1702 examples (the notation in the latter one is different from the modern one, but it seems that the technique is the same).
Unfortunately prior to the turn of the 18th entury the math literature was written and published mainly in Latin, which I don't know. However, a single 1673 source on a related topic which doesn't appear to explicitly state this rule refers to 10th Book of Euclid's Elements, is the method actually as early as that? I tried to read Euclid as well as two modern retellings, but didn't find a formulation of this technique: the classic seems to describe related properties of irrational expressions rather than instructing how to solve problems here.