Do you need to get permission to publish a quote or small excerpt?
Citations are how the academic world works. Any copyright law that I know of respects this system, and covers quotations under "fair use". Also, academics love to get cited - in the proper way. Presumably, they wrote the original source with the hope that they will get cited. Asking if they agree their text to be cited is as counterproductive as asking a supermarket employee if the peaches in the produce section, with the price tag on them, are meant for sale. 

You still have to make sure that it is what academics understand as a "citation". That is, if you make a claim in your own words, you still mark it with a reference. If you reuse a snippet, you put it in quotation marks and mark with a reference. The boundaries between "a phrase" (you don't have to attribute it, because it's too short), "a snippet" (it is a quotation, you may use it as long as you attribute it) and "too long" (you reused too much to count as a quotation) are fluid. They depend on the actual length of the text copied, on the length of your text, and on the function of the text in both the original work and yours. Still, with something like 1-3 sentences (attributed!) you are typically on the safe side. 

Images are a different beast. They don't "work" like quotations. Even if the author is not interested in pursuing copyright for them, their publisher may well do it. I don't know why this is the case, it may be because images are more salient than sentences, because they frequently contain too much information (often representing the core of somebody's theory), or because it used to be much more work to create an image than to type text. Anyway, the appropriate thing is to always get a permission to reprint images. You can even place a notice in the caption, for example "A concept for creating world piece, from [Smartass2016]. Reprinted with permission". 

Bottom line, if you quote a sentence or two, that's OK, just add a reference and it is a proper citation. If it is an image, ask for permission. If it is something else, best to contact your publisher and ask for advice.