Leader of opposition, Gulab Chand Kataria, said the whip was issued to the party MLAs asking them to remain present during the session but the four left the assembly.
Replying to the debate on the motion, Chief Minister Ashok Gehlot criticised the BJP, accusing it again of trying to bring down his government.
Deputy Leader of the Opposition Rajendra Rathore mentioned Pilot's name in various matters, including the recent political developments in the state and a SOG notice issued to him.
Floor test HIGHLIGHTS: A very good majority, says Pilot after Gehlot-led Rajasthan govt wins trust vote in state assembly
In a House of 200, the Congress party has 107 MLAs and the support of Independents and allies.
It will be victory of truth, says Gehlot as Congress government to go for trust vote in Rajasthan Assembly
The session comes after a month-long political crisis in the state which started with the rebellion by Sachin Pilot and 18 other Congress MLAs.
They said the high court cannot exercise the power of judicial review in 'purely political questions' but can examine questions if they emanate from a constitutional duty or obligation.
Senior spokesperson of the party Ajay Maken said it was "status quo" as far as the rebel Congress MLAs in Rajasthan are concerned.
Earlier, the governor had rejected three proposals by the Gehlot government seeking Assembly session from July 31.
"Happy Birthday and best wishes to the Rajasthan Legislative Assembly Speaker Dr CP Joshiji. I pray to God for your good health and long life," Pilot tweeted.
The Rajasthan Assembly saw a ruckus on Thursday over the bus accident in Bundi district, which resulted in the death of 24 people.
Kerala and Punjab state legislative Assemblies in the past have only passed resolutions against CAA, which made Rajasthan the first to implement against all three.
Countless swarms of locusts from Pakistan have caused extensive damage to crops, wreaking havoc for farmers in western Rajasthan.