Rangers v Hamilton Academical will be played on Saturday 4 March and will kick-off at 12.30pm.
The Ibrox club, beaten finalists last season, booked their place in the last eight with a narrow 2-1 win over Morton on Sunday.
Hamilton squeezed through against Dunfermline, beating the Fife side on penalties in a replay on Tuesday night.
Celtic’s quarter-final against St Mirren has been moved to Sunday 5 March and will also kick-off at 12.30pm.
The Scottish champions are on course for a domestic treble this season and are seeking to reach their first Scottish Cup final since 2013.
St Mirren, involved in a relegation dogfight in the Championship, defeated Celtic in the semi-finals of the League Cup that year.
The Aberdeen v Partick Thistle quarter-final will also be shown live, with BBC Scotland screening the game at Pittodrie on 5 March (kick-off 3pm).
The only last-eight tie which has not been selected for live television coverage is the game between Hibs or Hearts and Ayr United.
The Edinburgh clubs are due to meet in a fifth-round replay at Easter Road next Wednesday.