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I want to quote a text in which re of the word relationships is on one line while lationships is on the next, and a hyphen follows re (re-) to indicate that the word continues on the next line. When I want to quote the text, can I simply write down the word relationships or must I write down re-lationships?

I'm interested in what the Chicago Manual of Style considers correct.

Except in poetry, which retains its line breaks even when put on a single line, it doesn't make sense to retain a line end hyphen elsewhere on the line. It just wouldn't make sense and it would look weird. Chicago says this specifically on their website:

A hard hyphen is one that is typed deliberately and that must remain whether the phrase falls at the end of a line or in the middle of a sentence. An example is the hyphen in “two-thirds.” A soft hyphen, on the other hand, is there only when a word must be broken at the end of a line; it has likely been inserted by a word-processing feature, and it should disappear if the word falls in the middle of a sentence during typesetting.

While it's very rare, some books have a guide telling you which hyphens should be kept despite appearing at the end of the line (i.e. hard hyphens). As hyphenated expressions become older they often become single, non-hyphenated words, so it's not always obvious what hyphens are hard and what are soft. Look at the rest of the text to see if a word is hyphenated. Looking at other editions can also be helpful.

According to The Chicago Manual of Style (17th ed.), 2.96:

End-of-line hyphens should be marked to distinguish between soft (i.e., conditional or optional) and hard hyphens. Soft hyphens are those hyphens that are invoked only to break a word at the end of a line; hard hyphens are permanent (such as those in cul-de-sac) and must remain no matter where the hyphenated word or term appears.

Although this doesn't explicitly address the question, I think it's as specific as the manual gets.

In point of fact, when any author writes their text, they often do not insert soft hyphens. It's only once the manuscript is typeset that, according to the publishing style, soft hyphens are used in order to close up spacing and avoid the appearance of white space. If there were grounds for authors calling foul, any author whose book has been published with the use of soft hyphens would be able to object to the misrepresentation. But this never happens.

Some stylistic devices do need to be part of a quote—or their absence mentioned. These include such things as italics, bolded text, superscripts and subscripts. But those are explicit devices used by the author to distinguish one part of text from another. When quoting their text, those same devices should be employed in the quotation so that the distinguishing features are preserved.

However, other stylistic devices that we read in books have most likely not been chosen by the author, and they are not required to convey the meaning of what they have written. These include such things as the overall font face, font size, font colour, the margins and line spacing used—and soft hyphens.

In fact, if you think about it, if you were to copy text that included a soft hyphen, and then pasted it so that the text all appeared on a single line, the hyphen that you previously saw breaking a word between lines of text would disappear. Otherwise, what you are doing is copying what should be a soft hyphen and pasting it as a hard hyphen. So, while you don't need to copy soft hyphens, you certainly shouldn't be copying them as hard hyphens.