What's the name of the rule of inference where we conclude $p$ implies $q$, given the premise $p$ and $q$? 
What's the name of the rule of inference where we conclude $p$ implies $q$, given the premise $p$ and $q$?

Premise: $p$ and $q$.
Conclusion: $p$ implies $q$ (or the converse).
What's the name of this rule of inference?
 A: Going from $p$ and $q$ to $p \to q$ is not a very interesting inference, and it has no name.
A more interesting inference is going from $q$ alone to $p \to q$. And that one is sometimes called 'Conditionalization'
Indeed, the fact that you can get to $p \to q$ from $q$ alone is exactly why going to $p \to q$ on the basis of $q$ and $p$ is uninteresting, as the $p$ turns out to be completely unncessary in this inference.  You might as well ask the name for:
$q \to r$
$q$
$p$
$\therefore r$
A: It has no name. It can be thought of as either part of the definition of logical implication or as a theorem derived from "first principles."
In many introductory textbooks, the following truth table effectively defines logical implication:

Source: https://www.erpelstolz.at/gateway/TruthTable.html
The first line tells us that if $P$ and $Q$ are true, then the implication  $P\to Q$ is also true.
If you are familiar with the basic methods of proof, you can prove the $P \land Q \to (P\to Q)$ using a form of natural deduction as follows (screenshot from my proof checker):

The other lines of the truth table, among other things, are similarly proven here.
