In 1955, playing for Downlands College in a match against Gregory Terrace at the Brisbane Oval, he scored 155 runs.

[citation needed] John Woodcock of The Times wrote: "I doubt if it is possible to hit a cricket ball any harder than Dexter did today.

[citation needed] He scored his Test best of 88 against Pakistan in Australia the following season but was unavailable to tour the West Indies in 1965 for personal reasons.

He had a difficult 1965-66 Ashes series back home against England, taking just four wickets and one half-century.

Veivers was the non-striker when Simpson reached the 300 and, one Test later, when Fred Trueman took his 300th wicket by dismissing Neil Hawke.

[citation needed] Veivers was a Brisbane radio station executive before serving as the secretary of the Queensland Cricket Association from 1974 to 1977 and a state selector from 1977 to 1982.

[citation needed] He had a brief political career, holding the seat of Ashgrove in the Legislative Assembly of Queensland from 1983 to 1986 for the Labor Party.