The court hearing is one of the many attempts by opposition parties and civil organisations to have Zuma removed from office.
Zuma and National Assembly Speaker Baleka Mbete are cited as main respondents in the UDM application.
The UDM is supported by other opposition parties in the application. Last month, Mbete postponed the no-confidence motion, initially scheduled for 18 April, to allow the Constitutional Court to hear the UDM application and make a ruling.
The application for the no-confidence motion was brought by the Democratic Alliance (DA).
The UDM wants the highest court in the land to decide whether MPs can vote through a secret ballot.
The UDM application was prompted by Zuma's controversial removal of finance minister, Pravin Gordhan, and his deputy Mcebisi Jonas in a midnight cabinet reshuffle in March.
After Gordhan was fired the country's credit rating was downgraded to junk status by two rating agencies.
In its arguments before court, the UDM says the downgrade meant government has to find an extra R30 billion a year, or R300 billion over 10 years to service debt.
That money, argues the UDM, could instead be used for the benefit of South Africans, especially the poor, who depend on State social grants.
South Africa is facing an "urgent economic and political crisis due to the conduct of the President" which have begun to have real effects on ordinary South Africans, argues the UDM.
"It is thus hardly surprising that ordinary South Africans are angry, and that many are demanding the removal of Zuma as recently witnessed in countless 'Zuma Must Go' marches nationwide. One can only think how R300 billion could radically transform the lives of those who need it most."
The party further argues that the usual open vote would put ANC MPs who support the motion against Zuma in "an intolerable and unconstitutional position".
The UDM adds: "If they follow their conscience and vote for the motion of no confidence, they will certainly be disciplined and likely be expelled from the ANC, and lose their seats in Parliament."
"This will not merely have adverse consequences for them personally but will mean that the National Assembly and the public are deprived of their services and contribution going forward."
Zuma in his reply argues against the secret ballot, adding that MPs are required to carry out the decisions of the respective political parties.
"The applicant [UDM] must show that the Constitution requires a secret ballot to be cast in relation to a motion of no confidence in the President. This the applicant has not established, instead, the applicant admits that the Constitution is silent on how that vote should be cast," argues Zuma.
"I am advised therefore that the applicant has not made a case to invoke the exclusive jurisdiction of the court, and the application stands for that reason to be struck off the roll."
Meanwhile Mbete says although she was open to persuasions by the opposition parties to consider a secret ballot, her difficulty is that neither the Constitution and the rules of Parliament make provision such a move.
Organisations such as the Council for the Advancement of the South African Constitution, the Unemployed People's Movement, Institute for Security Studies and the Shosholoza Progressive Movement have joined the hearing as friends of the court.