If this question has an answer, the most likely answer is "No, they don't". But in my opinion, the question is not really meaningful. As far as I know, things like the [ej] in "ejus" were not traditionally categorized as diphthongs (I’m not entirely sure though).
The answer is "No" from at least one modern perspective. If anything is a diphthong in Latin, "au" in words like cautus is. "Au" in this position was traditionally classified as a diphthong, and even though there are some indications of potential monophthongization to a long mid back vowel (e.g. cauda > coda) that change did not run to completion in the ancestor of all the Romance languages, so it’s pretty safe to say that “au” was not realized as monophthong in Classical Latin.
But some sources such as Cser 2016 say that even "au" in pre-consonantal position was not a diphthong, but rather the same vowel + consonant sequence /aw/ that can be found before a vowel in forms like căveo (p. 35). Cser analyzes Latin as having no phonemic diphthongs of any type.
Even if you disagree with Cser and think of au, ae and oe as single vowel phonemes /a͡u/, /a͡e/ and /o͡e/ that constitute a nucleus (analogous to long vowel phonemes like /aː/, /iː/, /uː/), there are some etymological arguments for thinking that [jː] in words like major, cujus and ejus was a geminate consonant /j.j/ (with the first "part" occupying the coda of the preceding syllable, the second "part" occupying the onset of the following syllable, and no part in the nucleus of any syllable). In Italian, the regular reflex seems to be the long consonant /ʤ.ʤ/ (note: phonetically, there is only one release, so something like [dːʒ], but the phonological representation seems to be generally thought of as geminate /ʤ/). The Latin sequence /j.j/ sometimes represented an older sequence of some other consonant + /j/: e.g. in major, /j.j/ comes from *gj.