Why we treat sentence letter $Q$ as conclusion in one form and premise in other?
I completely agree with others that the terms "premise" and "conclusion" don't apply to things that aren't implications. By analogy, in $P\land Q$, $P$ and $Q$ are called "conjuncts", but it would make no sense to ask what the conjuncts of $P\to Q$ are.

However, there is terminology that could make sense of the connection that is more commonly used in type theory than logic but is just as applicable to logic. We can talk about "positive" and "negative" positions. In $P\to Q$, the premise, $P$, is in negative position and the conclusion is in positive position. These senses "multiply" in that in $(R\to S)\to Q$, $S$ is in negative position, but $R$ is in positive position (because it's in a negative position within a formula that is itself in negative position). Since $\neg P\equiv (P\to \bot)$, ($\bot$ is the always false formula) negation effectively swaps negative and positive. In $P\land Q$ and $P\lor Q$, both positions are positive. (The general intuition is that propositions in positive positions are possibly asserted, while propositions in negative position are possibly being assumed. They might not actually be asserted/assumed, e.g. in $P\lor Q$.)

This positive/negative notion can be a quick sanity check for equivalences and can help guide proof search. (Negative propositions will likely need to be assumed at some point and positive ones proven. Also, particularly for purely logical theorems, you are often trying to bring negative and positive occurrences together, e.g. as in $P\to P$.) For your equivalence $$P\to Q \equiv \neg (P\land\neg Q)$$ we can see on the right-hand side that $P$ occurs in a positive position within a formula in negative position and so is negative, while $Q$ occurs in negative position within a formula in negative position and so is positive. This matches the left hand side.