The drop means the Mercedes driver, second in the championship behind Ferrari's Sebastian Vettel, cannot equal Michael Schumacher's pole position record of 68 at his home British Grand Prix next week.
The triple world champion, who is currently on 66 poles and won in Austria last year, will have to wait until Hungary at the end of the month at the earliest.
He had been favourite to take the top slot at the scenic Red Bull Ring after dominating Friday's two free practice sessions with Vettel second fastest.
Qualifying is on Saturday after a final practice and Hamilton will be hoping that he is fastest to minimise the drop and that Finnish team mate Valtteri Bottas is second to deny Vettel the top slot.
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A Mercedes team spokesman said the gearbox problem resulted from the previous race in Azerbaijan, where Hamilton started on pole but finished fifth after having to pit to fix a loose headrest.
Formula One regulations stipulate that gearboxes must last six races unless a driver fails to finish for whatever reason.
The Formula One technical delegate said in a statement he had been informed by Mercedes last Tuesday that the gearbox was damaged and needed changing, a detail the team had kept secret.
Hamilton, who had told reporters earlier that pole was the goal, is 14 points behind Vettel after eight of 20 races. Both drivers have won three races.
The pair clashed in Baku when Vettel drove into the back of the Mercedes while they were behind the safety car and then banged wheels angrily after pulling alongside.
Vettel, who finished fourth in Baku, apologised for the incident of "road rage" -- and for claiming that Hamilton 'brake-tested' him -- after a hearing in Paris on Monday.
Mercedes ruled out that contact as the cause of the gearbox problems.
(Reporting by Alan Baldwin, editing by Ken Ferris/Peter Rutherford)
