There is this Dragon Ball family tree been shared in Dragon Ball related websites, which shows the name "Tarble" (ターブル - Tāburu)) in japanese. Is this official? Has Tarble been canonized? IIRC a brother of Vegita was mentioned in Battle of Gods but nobody said his name
Tarble has been confirmed to be canon on the source you included in your question. You can view the text/ translation with regard to the same here.
As far as the official Dragon Ball continuity is concerned, Tarble is non-canon.
Tarble is mentioned in the movie adaptation of Dragon Ball Z: Battle of Gods, but was not mentioned at all in the episodes of Dragon Ball Super that covered the events of the movie.
Whether or not he will be mentioned in the new movie, Dragon Ball Super: Broly, remains to be seen. The movie officially releases in Japan on December 14th.