The prime factorization of 12 is
12 = 2²×3
To find denominators suitable for your purpose, you need to group the factors
{1, 2, 2, 3}
into two groups with different products that share at least one of the numbers that is not 1.
For example, your grouping could be (2, [2), 3], where a 2 is shared. That is, your denominators could be 2*2=4 and 2*3=6.
Another possible grouping is (1, [2), 2, 3], giving rise to denominators of 2 and 12.
Yet another is (1, [3), 2, 2], for denominators of 3 and 12.
Using the first grouping, a pair of mixed numbers matching your requirement might be
1 3/4, 5 1/6