[noun] + めく [noun] + めいた + [noun] [noun] + めいている
めく turns a noun into a verb meaning "to take on a touch of", "to have an air of": something is not quite that thing but gives off its feel. It goes straight after the noun and conjugates as a う-verb, so the past is めいた and the て-form めいて.
Before a noun, the form is めいた, and that is where you meet めく most often: なぞ謎めいた (mysterious), ひにく皮肉めいた (with a sarcastic edge), せっきょう説教めいた (preachy), じょうだん冗談めいた (half-joking).
With a season, めく says the air is starting to feel like it. はる春めく is the most common, and 秋めく and 夏めく follow the same pattern. It usually comes with きごう〜てくる for the change.
めく belongs to writing and careful speech, and it takes only a limited set of nouns: seasons, and words for a kind of talk or manner such as 謎, 皮肉, 説教, 冗談, 言い訳, 芝居. In conversation, and for most other nouns, the everyday word is っぽい.