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 a unknown Gnostic type of genealogies that include angels. However there seems to be strong support that it was a particularly Jewish disturbance being spoken of in Timothy and Titus, 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 in 1 Timothy there are references to Jewish Law in verses 7-19, 2 Timothy 4:4 and also in Titus 1:10, 1:14, 3:9. Furthermore in Gnostic literature the concept of angelic hierarchies and emanations are not presented as 'genealogies' per say.
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.