I would like to know how to compute multiplication of two valid EC points over a curve E with generator G.
i.e. Given only P and Q points then how to compute R = P * Q
where $P = p G$, $Q = q G$ and $R = (p \cdot q)G$.
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There is no standard "multiply two group elements" operation in an additive group. So you first need to define what you mean by $P*Q$. From the comments I gather that you want $P*Q = q P = p Q = (p \cdot q) G$.
The computational Diffie-Hellman (CDH) problem is:
Given $P=pG$ and $Q=qG$ compute $(p\cdot q)G$.
which is clearly equivalent to your problem.
If the discrete logarithm (DL) problem is easy for the group, you can first solve $P=pG$ for $p$ and then compute $P*Q=pQ$. But on the curves we use in cryptography both DL and CDH are believed to be hard. While such groups may exist, I'm not aware of any groups in which DL is hard but CDH is easy.
Another problem of interest is the decisional Diffie-Hellman problem:
Given three group elements $P=pG$ and $Q=qG$ and $R$ decide if $R = (p \cdot q) G$.
There are some curves for which the decisional Diffie-Hellman problem is easy while the computation Diffie-Hellman problem and the discrete logarithm problem are hard. These groups are know as gap-groups and are useful in cryptography, for example they're used in the BLS signature scheme.
Well the equation $R = P * Q$ simply isn't possible on an elliptic curve. The group of points on the EC is an additive group. Meaning it is only possible to compute $P + Q$ or $[m] P$ for some integer m. Taken $P=p \cdot G$ and $Q=q \cdot G$ you already got the answer yourself: $R=(p \cdot q)G$. Simply add the point $G$ to itself $(p \cdot q)$-times.