Have you stopped beating your wife?
implies that you have been beating your wife, and
Was it Paul or Peter who opened the gate?
implies that either Paul or Peter opened the gate. More controversially, but also plausibly, assertions can imply questions, e.g., 'Either Paul or Peter opened the gate' naturally raises the question, "Which did it, Paul or Peter?", and this question-raising is either implication or something very like it.
But, as with imperatives there are a great many complications. For instance, questions do not seem to be the sort of thing that can be true or false, and this complicates practically everything. Interestingly, not everyone has agreed that questions can be neither true nor false -- Bolzano is the most famous case, since he argued that every question is actually an assertion about the one asking it. There is a certain plausibility for this with regard to some kinds of questions. For instance, if I ask a question in order to learn something I did not know, e.g., "What is the dominant predator on the island of Malta?", then this is not far from saying, "I do not know the dominant predator on the island of Malta." Indeed, the latter, in some circumstances, might be taken as an implicit question. And it is notable that much of the more interesting recent work on questions has tended to link it to various kinds of epistemic modal logics.
Nonetheless, Husserl showed that Bolzano's position is quite problematic, pointing out (among other things) the absurdity of thinking that when we are silently wondering about something we are doing the same thing as sitting there asserting our lack of knowledge about things. At the very least, some questions are not assertions, and for many questions it is absurd to take them to be assertions about the questioner.
Yet it's not so clear that Bolzano was wholly off-base. One of the things we do with questions is soften assertions. For instance, in Vietnamese, nhé? (or in some dialects, nha?) turns a statement into a suggestion, along the lines of how we might say in English, "I'll take you home now, ok?" This is arguably an assertion -- I am actually saying that I'll take you home, I'm just letting you have a say in the matter if you object. It is also prima facie a question. There seems to be a spread of ways in which this can work. In Vietnamese, if I understand correctly, Tôi đi nhé?, "I am leaving, ok?" is a different kind of question from Tôi đi à?, which might be something more like "I'm leaving!?" or "Am I leaving!?" and both from Tôi đi không?, "I'm leaving, aren't I?" or "Am I leaving?" (One could perhaps argue that these are actually compound, with an assertion part and a question part, but when we look at these things in use, it's not clear that this sort of analysis actually sheds any light on the meaning.)
One variation would be to take questions to be to assertion as incomplete to complete. So, for instance, an assertion might be, "John went to the party last night." Some corresponding questions might be:
Did John go to the party last night?
Who went to the party last night?
What did John do?
Who did what?
In each case we are missing something. If we supply the answer, we get something equivalent to the original assertion. For instance, "Did John go to the party last night? Yes", is equivalent to "John went to the party last night", and so is "Who went to the party last night? John."
There are appropriate and inappropriate answers to questions. For instance, if you ask, "Who is in the house?" and I reply, "Slowly," my answer is not just incorrect, it's not even the right kind of answer to be correct. Thus there is a longstanding tendency to try to account of questions in terms of their possible answers. Thus Hamblin influentially argued that questions create a situation in which we choose among possible answers, which he held to be propositions. A potential strength of this is that it makes the logic easier -- you could then say a lot about the logic of questions based on the logic of the assertions that make up their possible answers. One difficulty is that the appropriate answer to a question does not always seem to be a proposition. If I say, "What color is John's shirt?", I only need an answer like "Red", I don't need, "John's shirt is red." I could certainly say the latter, but the only part of it that is doing any work answering is the term 'red', and it seems that instead of thinking in terms of propositions, we can just take the answer to be supplying what's needed to finish making the proposition -- which we can do either by just supplying the particular element needed or by giving the finished product with the element supplied. An even more serious difficulty is that the possible answers to a question seem to have to be severable from understanding the question itself. We can make sense of a question without having much idea as to what its possible answers really are, so we don't want to characterize questions as if they could only be understood if you knew all the ways they could be answered. We need to be able to know what would count as a possible answer without having to know the possible answers.