Drew Brees had the most games by a quarterback with 5+ passing touchdowns and 300+ passing yards, with 9 games.

which qb has the most career games with 5 passing tds and 300 yards passing
Interpreted as:
Which qb has the most career regular-season games with 5 or more passing tds and 300 or more yards passing?
BreesBradyManningMarinoRoethlisbergerRodgersStaffordBlandaMahomesWilsonMoonKellyMontanaJohnsonJurgensenNewtonRomoMcNabbCulpepperFavreFergusonMeredithTittle97764444333333322222222
NAMEGPTDYDSCMPATTPCTAVGYDS/GTD%INTINT%SCKSCKYRATE
Drew BreesDrew Brees9483,42525532977.510.4380.614.641.2838144.6
Tom BradyTom Brady7372,59619726075.810.0370.914.210.41172144.8
Peyton ManningPeyton Manning7382,70417125766.510.5386.314.872.7534129.6
Dan MarinoDan Marino6312,33315024561.29.5388.812.752.0534123.9
Ben RoethlisbergerBen Roethlisberger4221,49010913879.010.8372.515.900.0638151.2
Aaron RodgersAaron Rodgers4221,4159113070.010.9353.816.900.0426145.4
Matthew StaffordMatthew Stafford4201,61411717666.59.2403.511.463.4848119.4
George BlandaGeorge Blanda4221,3647413455.210.2341.016.453.7114.6
Patrick MahomesPatrick Mahomes3171,2208711675.010.5406.714.732.6434137.2
Russell WilsonRussell Wilson3151,0387711368.19.2346.013.300.0960136.7
Warren MoonWarren Moon3151,1007310967.010.1366.713.832.8538128.1
Jim KellyJim Kelly3161,118669966.711.3372.716.266.1546119.0
Joe MontanaJoe Montana3161,3339414067.19.5444.311.432.11386126.9
Charley JohnsonCharley Johnson317986448353.011.9328.720.556.0110.2
Sonny JurgensenSonny Jurgensen3151,155539058.912.8385.016.777.8110.4
Cam NewtonCam Newton210671538661.67.8335.511.611.2440119.9
Tony RomoTony Romo210812476572.312.5406.015.411.5544147.6
Donovan McNabbDonovan McNabb210806557276.411.2403.013.900.0630152.0
Daunte CulpepperDaunte Culpepper210821628771.39.4410.511.522.3516129.5
Brett FavreBrett Favre210724527866.79.3362.012.833.8533119.9
Joe FergusonJoe Ferguson210786578567.19.2393.011.811.2130.8
Don MeredithDon Meredith210752335066.015.0376.020.000.0148.8
Y.A. TittleY.A. Tittle213846488159.310.4423.016.022.5124.3
Daniel JonesDaniel Jones15352284266.78.4352.011.900.016132.1
Deshaun WatsonDeshaun Watson15426283384.812.9426.015.200.000158.3
StatMuse has game-level data for passing touchdowns and passing yards going back to the 1950 season.

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