Drew Brees had the most games with 5+ passing touchdowns, with 11 games.

most games with at least 5 touchdown passes
Interpreted as:
most regular-season games by a player with at least 5 touchdown passes
BreesBradyManningRoethlisbergerMarinoWilsonRodgersBlandaMahomesStaffordFavreMoonJacksonWatsonNewtonCulpepperKellyMontanaKriegFoutsMeredithDawsonJohnson119966555444433333333333
NAMEGPTDCMPATTPCTYDSAVGYDS/GTD%INTINT%SCKSCKYRATE
Drew BreesDrew Brees115829638177.73,96310.4360.315.241.01053145.2
Tom BradyTom Brady94725133774.53,2839.7364.813.910.316104143.1
Peyton ManningPeyton Manning94921931569.53,23010.3358.915.672.2643133.1
Ben RoethlisbergerBen Roethlisberger63214618877.71,92710.3321.217.010.51064146.7
Dan MarinoDan Marino63115024561.22,3339.5388.812.752.0534123.9
Russell WilsonRussell Wilson52512117369.91,6189.4323.614.510.61173136.5
Aaron RodgersAaron Rodgers52711215970.41,69810.7339.617.000.0426144.9
George BlandaGeorge Blanda5288715855.11,5549.8310.817.763.8112.7
Patrick MahomesPatrick Mahomes42211114676.01,48710.2371.815.142.7541136.0
Matthew StaffordMatthew Stafford42011717666.51,6149.2403.511.463.4848119.4
Brett FavreBrett Favre4219414365.71,2798.9319.814.764.2947116.2
Warren MoonWarren Moon4209414166.71,3169.3329.014.253.5538121.3
Lamar JacksonLamar Jackson315476374.670511.2235.023.800.0410150.5
Deshaun WatsonDeshaun Watson315608471.492611.0308.717.900.0313147.1
Cam NewtonCam Newton3157412061.79177.6305.712.510.8660121.4
Daunte CulpepperDaunte Culpepper3157911071.81,0639.7354.313.621.8716134.2
Jim KellyJim Kelly316669966.71,11811.3372.716.266.1546119.0
Joe MontanaJoe Montana3169414067.11,3339.5444.311.432.11386126.9
Dave KriegDave Krieg315659866.38218.4273.715.333.1435119.1
Dan FoutsDan Fouts3167912264.81,0388.5346.013.121.6322124.2
Don MeredithDon Meredith315487266.798313.7327.720.811.4143.5
Len DawsonLen Dawson316569360.297610.5325.317.244.3117.7
Charley JohnsonCharley Johnson317448353.098611.9328.720.556.0110.2
Sonny JurgensenSonny Jurgensen315539058.91,15512.8385.016.777.8110.4
Jameis WinstonJameis Winston210334967.33948.0197.020.400.018131.3
StatMuse has game-level data for passing touchdowns going back to the 1950 season.