One of the cubes, say, B, should have digits 0 to 5, and the other, say A, should have digits 0, 1, 2, 6, 7, 8.

With this,
Dates 1 to 5 can be represented with 0 from cube A and 1 to 5 from cube B;
Dates 6 to 8 can be represented with 0 from cube B and 6, 7, 8 from cube A;
Date 9 can be represented through a ‘trick’ – using 0 of Cube B and 6 from Cube A turned ‘upside down’.
Date 10 can be shown with 1 and 0 from either cube;
Date 11 can be made of 1s from both cubes;
Date 12 to 15 can be made from 1 of cube A and 2 to 5 of cube B;
Dates 16 – 19 can be represented similar to dates 6 – 9 by replacing the 0 with 1 from cube B;
Dates 20, 21: 2 and 0 / 1 from either cube;
Dates 22 – 25 can be made similar to dates 12 to 15, replacing the 1 with 2 from cube B;
Dates 26 – 29 can be done similar to 16 – 19;
Dates 30 and 31: Use 3 from cube  B and 0 and 1 of cube A.