How do journal editors and reviewers view previously published conference paper? I am submitting my paper to a journal, but a very small part of the work was previously presented in a quite prestigious international conference (although not the top). The journal policy allows for this, given the revised journal version has at least 30% novelty as a thumb rule. It even allows some copied texts and figures, which I do not have. 

So from what I see, it is not absolutely necessary for me to mention the conference edition as part of this work although I am citing it as relevant, along with other references. But researchers will understand that it is not black and white, but gray. The previous research will always somewhat influence the future research. 

If I do mention the conference presentation as part of the submitted work, will it make the work more authentic to the reviewers? Will they somehow take it more seriously, so to say? I am interested in the opinions of experienced reviewers and editors. 

Just to make it clear, in my field, journals are considered more prestigious and they publish the more refined, expanded versions. 

Also, this journal has an anonymous review process, but not double-blind.
I do not think you really have a choice here: if part of the paper has been previously published elsewhere, you need to mention that fact explicitly.  Not mentioning such is generally considered self-plagiarism and looked on very badly.

Now, depending on how you are re-using the material, there are two different ways that you might do it.  If you were doing a typical "extended version" paper, then the appropriate thing to do is to state this fact both in the introduction and in the cover letter.  

It sounds, however, like it's less an extended paper than a repeat of a particular section.  In that case, I would instead recommend putting early in the repeated section a statement like:


  "This section has been adapted from [citation]"


For example, I often do this when presenting background sections, on which the new results of a paper will then be built.  The background section will be adapted to fit the circumstances, of course, but rather than artificially try to paraphrase definitions just for the sake of avoiding self-plagiarism, I prefer to handle the issue by making a clearly stated reuse of background material that is similar in spirit to a block quotation.