Pope was not out 183 and Smith, playing his maiden first-class match, was not out 123 with Surrey 389 for 4 at stumps on day two of the four-day match in reply to MCC's first-innings total of 265.
It was Pope's sixth first-class century in 26 matches, while Smith became the ninth Surrey player to score a ton on first-class debut. It followed an impressive performance by Smith behind the stumps on the opening day after he completed a stumping to dismiss opener Will Rhodes (46), caught Dan Lawrence - who was MCC's top scorer with 58 - and an excellent run out to dismiss Dom Bess for just four.
Surrey resumed on 20 without loss as a sandstorm hit the ICC Academy ground and openers Rory Burns (12) and Mark Stoneman (21) were both caught in the slips by Lawrence off the bowling of Sam Cook.
Scott Borthwick (16) was caught and bowled by MCC captain Stuart Broad for 16, while Will Jacks, who scored a spectacular 25-ball century in a pre-season T10 game against Lancashire last week, suffered the same fate at the hands of Peter Trego to be out for 20 in his latest knock.
The 21-year-old Pope, who played two Tests for England against India last year, and Smith proceeded to put on an unbroken 245-run partnership for the fifth wicket, with Pope hitting 19 fours and two sixes en route to passing his previous first-class best of 158 not out. Smith brought up his century with consecutive fours off Bess - he hit 15 fours and two sixes in all.