Tom Brady has the most games by a quarterback with 1+ passing touchdown, with 266 games.

which qb has the most 1 td games
Interpreted as:
Which qb has the most 1 or more passing td regular-season games?
BradyBreesFavreManningRiversRoethlisbergerMarinoManningTarkentonRyanRodgersTestaverdeElwayMoonPalmerUnitasStaffordBledsoeKriegHadlHartFlaccoMontana266250249233209205203193187186179170166165152146144144142139137133132
NAMEGPTDCMPATTPCTYDSAVGYDS/GTD%INTINT%SCKSCKYRATE
Tom BradyTom Brady2666026,2359,59165.073,2617.6275.46.31651.74432,770101.8
Drew BreesDrew Brees2505716,4199,37468.573,1827.8292.76.11962.13582,522103.3
Brett FavreBrett Favre2495085,3618,50663.062,3877.3250.66.02563.04262,76192.5
Peyton ManningPeyton Manning2335395,5488,40966.065,9697.8283.16.42222.62691,782100.1
Philip RiversPhilip Rivers2094214,6217,01465.956,3078.0269.46.01692.43892,347100.4
Ben RoethlisbergerBen Roethlisberger2054034,5627,00865.154,9347.8268.05.81632.34212,81898.5
Dan MarinoDan Marino2034204,3217,16460.354,0487.5266.25.92112.92261,63391.1
Eli ManningEli Manning1933664,1426,79161.049,0237.2254.05.41992.93162,23288.7
Fran TarkentonFran Tarkenton1873423,0055,10858.839,6967.8212.36.72013.989.4
Matt RyanMatt Ryan1863594,4926,79066.251,5077.6276.95.31382.03432,26398.0
Aaron RodgersAaron Rodgers1794274,1276,26465.949,7647.9278.06.8821.34242,817107.4
Vinny TestaverdeVinny Testaverde1702753,0745,32157.838,3007.2225.35.22114.03142,26480.9
John ElwayJohn Elway1663003,1245,30258.940,3817.6243.35.71482.83482,64890.1
Warren MoonWarren Moon1652913,3645,65959.442,3087.5256.45.11863.33442,55986.2
Carson PalmerCarson Palmer1522943,4315,39163.640,5477.5266.85.51592.92741,87192.3
Johnny UnitasJohnny Unitas1462902,2013,96355.532,7278.3224.27.31814.688.1
Matthew StaffordMatthew Stafford1443013,4835,47763.640,9707.5284.55.51262.33212,11195.0
Drew BledsoeDrew Bledsoe1442512,9995,10358.835,7927.0248.64.91382.73342,24285.4
Dave KriegDave Krieg1422612,4224,05159.830,7307.6216.46.41413.53422,68490.5
John HadlJohn Hadl1392441,8713,60551.927,9477.8201.16.81774.979.7
Jim HartJim Hart1372091,9583,73352.527,4487.4200.45.61794.875.1
Joe FlaccoJoe Flacco1332242,9504,71562.632,6846.9245.74.81052.22691,91789.7
Joe MontanaJoe Montana1322732,6794,18564.032,9177.9249.46.51072.699.3
Russell WilsonRussell Wilson1312772,5943,93166.031,7168.1242.17.0661.73592,364107.2
Dan FoutsDan Fouts1302542,6384,42259.735,2998.0271.55.71804.187.2
StatMuse has game-level data for passing touchdowns going back to the 1950 season.