For solid-liquid mixtures, cooling crystallization process can be used to separate the solute substances from each other. Assume that we have two salts that have very close solubility in a certain solvent which we are going to use. First, we dissolve the salts in the solvent while heating. After a point, we let the mixture cool down and observe the crystallization. My question is, how can we do this experiment accurately if they have really close crystallization point that they would nearly crystallize at the same time?
Under the right conditions, crystallization can be a very effective means of obtaining a pure solid form of the solute. Unfortunately, in your case one of those conditions is violated; you have a second solute with similar solubility characteristics. You didn't quantify the solubilities of the solutes, but the phrase "very close solubility in a certain solvent" tells me that under the conditions of your system, crystallization will not be an effective means of obtaining a pure solid form of the solute.
One option, as mentioned in a comment to your question, is to choose a different solvent. Selecting the right solvent / solvent mix can be a bit of an art and will also require some research and experimentation on your part. Because you didn't discuss the specific solvents or solutes used in your system, it is not really possible to give more specific advice on solvent selection.
Another option is to choose a different means of solute separation, like some type of chromatography. Again, not knowing anything about your compounds makes it difficult/impossible to give further advice regarding this technique.