In both 1 Timothy 1:4 and Titus 3:9 Paul warns his students against "endless genealogies." Obviously the Bible itself contains genealogies, even in the Gospels, so it seems like there must be something more going on than just people are reading these portions of Scripture - maybe a particular way of reading them? Do we know anything about the practices which the letters are warning against?
Most commentators circle around the ideas that these are either Jewish genealogies or an unknown Gnostic type of genealogies that include angels. However there seems to be strong support that the pastorals speak of a particularly Jewish disturbance, as compared to say Colossians which may have been related to a mystic type of Jewish Gnosticism, somewhat like the Essenes.
The argument is made this way because there are references to Jewish Law in 1 Timothy 1:7-19; 2 Timothy 4:4; and Titus 1:10, 1:14, and 3:9. Furthermore in Gnostic literature the concept of angelic hierarchies and emanations are not presented as 'genealogies' per se.
Therefore assuming this was a purely Jewish debate about genealogies, possibly by the Judaisers, it was probably not related to arguing against the lineage of the Messiah, but about meaningless matters. Possibly certain people claimed to be superior or more blessed based on controversial beliefs about certain ancestries along Jewish folklore? There is really no way of knowing for sure as these precise debates do not seem to be extant in any writings.
Paul is warning his Churches of fake genealogies such as those in Luke and Matthew. Following the destruction of the Temple if AD70 in which the official family genealogies of the Jews were burnt, the writers of Luke and Matthew (in trying to support claims Jesus was the prophesied Messiah) had to construct genealogies from whatever sources they could. Analytical assessment indicts substantial artistic license was used by both writers. EG according to prophesy the Messiah shall be from line of the House of David and rule over Israel but this would not fit if Jesus physical father was the Holy Spirit. Alternatively, Jeremiah prophesied that “No future king would follow from the issue of Jehoiachin” who was carted off by Nebuchadnezzar – in prophetic terms Joseph cannot be the father either. Evidence from Jesus birth and ministry, leading to his anointment and recognition by the Jews as the king-Messiah - strongly suggest Jesus contemporaries fully knew his family line and eligibility for Messiahship.