The concrete word to be used depends on the kind of the object you want to count and it is called a "classifier". There are hundreds of classifiers in Thai; much fewer classifiers than nouns, but still a lot of them. So you might associate each classifier with a class of nouns. If you cannot use resources in Thai (in the Thai script), you could perhaps sample the most common ones here.
Classifiers are in no way specific to questions; they are needed in any context where you need to count objects. So they will appear in "how many" type questions, in answers to those questions, and also in free standing sentences where you are counting anything. Instead of "five elephants", you'd have "elephant five animal". Chairs happen to belong to the same class like elephants, being multi-legged things, so "five chairs" would be "chair five animal". And so on.
Classifiers have additional grammatical functions in Thai as well. For example they can indicate definiteness; in this situation a classifier is used without a numeral, but its presence indicates that the noun is used in singular, so you might still interpret the classifier's presence to be involved in some form of counting.
One way of thinking about those noun classes could be to compare them to grammatical genders of languages like French or German or Russian, except that there are way more noun classes in Thai (compared to just three or so genders in those other languages, or to mere nine noun classes in Swahili). What French or German achieves with suffixes (on numerals and elsewhere), is achieved in Thai with standalone words. This is a very crude comparison which doesn't hold in every detail; I'm just trying to replace the notion of a "special word" by a richer parallel from languages closer to English.