Quarta has chaired the holding company since 2015, and became temporary executive chairman when Sir Martin Sorrell abruptly stepped down as chief executive last month.
But Glass Lewis said it had "severe reservations" involving the failure to publish the outcome of the investigation into personal misconduct that led to Sorrell’s resignation, The Guardian reports.
The result of this was that Sorrell was classified as a "good leaver", allowing him to keep up to £20m in future share payouts.
Shareholders should also vote down WPP’s pay report, Glass Lewis said, because of the lack of transparency around Sorrell’s departure.
Sorrell’s pay for 2017 fell more than 70% to £13.9m, with the then-chief executive receiving substantially less of his possible bonus than in previous years.
Glass Lewis said: "We harbour concerns as to the transparency and efficacy of the succession process. Despite previous assurances we believe the nomination committee has failed to adequately prepare for the replacement of Sir Martin.
"Our concerns are heightened by the opaque nature of the investigation into Sir Martin [and] his ‘good leaver’ status. Absent further information regarding Sir Martin’s retirement, we believe shareholders are unable to determine the extent to which he should be treated as a ‘good leaver’."
A WPP spokesman said that a vote against Quarta would be against the interests of investors at "an important time when the business needs stability until such time as a new chief executive is appointed".