Britain will also have two representatives in the semi-finals of the men’s doubles event.
Sixth seeds Neal Skupski and American Christian Harrison won 6-2 6-3 against Czech pair Petr Nouza and Patrik Rikl.
Luke Johnson and Polish partner Jan Zielinski were 7-6 (7-5) 6-2 winners over fourth seeds Marcelo Arevalo, of El Salvador, and Croatia’s Mate Pavic.
Skupski and Harrison will play third seeds Marcel Granollers of Spain and Argentina Horacio Zeballos for a final place, with Johnson and Zielinski taking on Australian wildcards Jason Kubler and Marc Polmans in the other half of the draw.
Andy Lapthorne and Gregory Slade were beaten in their respective quad singles first-round matches.
Lapthorne lost 6-4 7-5 to Turkish fourth seed Ahmet Kaplan, while Slade was beaten 6-1 6-3 by Brazilian qualifier Leandro Pena.
However, Lapthorne progressed to the quad doubles semi-finals alongside Australian Heath Davidson, achieving a 7-6 (7-1) 6-4 win against Chile’s Francisco Cayulef and Argentine Gonzalo Enrique Lazarte.