I have read this paper which is really old but it is nice paper. It says the following: "since there are exactly 30 distinct STS(7) on the same set, there are exactly 30 distinct SQS(8) on the same set". How he is calculated 30?
Also he says in next paragraph for 2520 distinct for SQS(10), how is it calculated?
Steiner Quadruple System - A Survey. see page 150-151.
Updated: I'm trying to give examples of STS(7) and count the number of possible designs. For example: if we begin with 123, then we have the following: 145,167,247,256,346,357. So depends on the first block we can always get different designs of STS(7). So to choose 3 numbers from 7 we have binomial coefficient (7 choose 3)=35. Still not correct; because in the survey paper he give 30.
As for the STS(7), the number $30$ is not too difficult to obtain, at least if you know that this design is unique up to isomorphism: In that case, every STS(7) on the set $\Gamma:=\{1, \ldots,7\}$ is obtained by applying a permutation $\alpha\in G:=Sym(\Gamma)$ to a fixed STS(7) (let's call it $\Sigma$).
On the other hand, two such permutations $\alpha,\beta\in G$ yield the same STS(7) if and only if they lie in the same coset of the group $H := Stab_G(\Sigma)$. It is not too difficult to show that $H\simeq Aut(\Sigma)$ and thus $|H|=168$.
Consequently, the number of distinct STS(7) on the set $\Gamma$ is $$|G/H|=\frac{|G|}{|H|}=\frac{7!}{168}=30.$$