When referring to the formal name of something in English, only the first letter is capitalized. (The exceptions include acronyms where each letter stands for a word: NYC, USA, NATO, UN.) Also, in the name of an article, the major words are capitalized and the small, connecting words (such as the, a, an, of, as, etc. are not). Quotation marks occur before the title of an article and after it. Moreover, a comma should be placed before the name of the article which is marked by quotation marks and after it. Thus, the sentence should read, Drummund Castle Gardens is the garden mentioned in the article, "Rest in the Park," in which a beech tree, planted by Queen Victoria in 1842, was preserved.
If I wanted to quote your first sentence and then include the name of the article in the quoted sentence, then the punctuation around the name of the article should be single quotation marks, not double ones.
"Drummond Castle Gardens is the garden mentioned in the article, 'Rest in the Park,' in which a beech tree, planted by Queen Victoria in 1842, was preserved."