How can you tell the difference between two organisms being heterozygous or homozygous without mating them?

it depends on the trait and the knowledge you have of its phenotype, for instance long hair in cats is a recessive gene, so a long haired cat would be homozygous.

the mating parts

We have to perform a cross called the TEST CROSS in order to differentiate between the one with homozygous dominant trait and one with heterozygous.

There would have to be some measurable difference, it is doing the measurements which can be tricky.

If you had plants with purple flowers, then the homozygous one would produce more pigment than the heterozygous one, even if they looked similar to the naked eye. So you could extract pigment from the two flowers and measure the difference by light absorption or something similar.

You could also look at the alleles using protein and/or DNA electrophoresis if you had the knowledge and technology to do so.

Most of the time it is just not possible to tell by inspection.