"Please recommend me a season with good weather."
Translation:날씨가 좋은 계절을 추천 해 주세요.
That way you would lose the connection between "good weather" and "season". Instead -가 makes the weather the subject of the main verb in your version: "The good weather recommends a season." (It can't be the subject of 좋은 because it's behind 좋은, and all arguments have to come before their verb).
Think of it this way: "the weather is good" as a full sentence is 날씨가 좋아요, right? You can turn describe a noun with that sentence by replacing the sentence-final ending with the atteibute one (-은 for adjectives) and keep everything else exactly like it is*: 날씨가 좋은 계절 "a the-weather-is-good season" = "a season with good weather". It's the same idea as with phrases 사람이 많은 장소 "a people-are-many place" = "a place with a lot of people".
I guess you could theoretically also say 좋은 날씨의 계절 "a season of good weather". But in practice it's usually more natural to use a verb phrase/sentence as the attribute if possible.
* With one caveat: Attribute clauses like this rarely if ever have their own topic (a noun marked with -은/-는). I think this is because the described noun is interpreted as the topic in the attribute clause. In other words, the full sentence corresponding to 날씨가 좋은 계절 would be (이) 계절은 날씨가 좋아요.